# Groups of order 8

This article gives information about, and links to more details on, groups of order 8

See pages on algebraic structures of order 8| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 8. See also more detailed information on specific subtopics through the links:

Information type | Page summarizing information for groups of order 8 |
---|---|

element structure (element orders, conjugacy classes, etc.) | element structure of groups of order 8 |

subgroup structure | subgroup structure of groups of order 8 |

linear representation theory | linear representation theory of groups of order 8 projective representation theory of groups of order 8 modular representation theory of groups of order 8 |

endomorphism structure, automorphism structure | endomorphism structure of groups of order 8 |

group cohomology | group cohomology of groups of order 8 |

## Statistics at a glance

To understand these in a broader context, see: groups of order 2^n|groups of prime-cube order

Since is a prime power, and prime power order implies nilpotent, all groups of this order are nilpotent groups.

Quantity | Value | Explanation |
---|---|---|

Total number of groups | 5 | See groups of prime-cube order, classification of groups of prime-cube order |

Number of abelian groups | 3 | equals the number of unordered integer partitions of 3, the exponent part in . See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |

Number of groups of class exactly two |
2 |

## The list

To learn more about how tocome upwith the list and prove that it is exhaustive (i.e., that these are precisely the isomorphism classes of groups of order 8), see classification of groups of prime-cube order

Common name for group | Second part of GAP ID (GAP ID is (8,second part)) | Hall-Senior number | Hall-Senior symbol | Nilpotency class | Minimum size of generating set | Probability in cohomology tree probability distribution |
---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 1 | 1/4 | |

direct product of Z4 and Z2 | 2 | 2 | 1 | 2 | 7/16 | |

dihedral group:D8 | 3 | 4 | 2 | 2 | 3/16 | |

quaternion group | 4 | 5 | 2 | 2 | 1/16 | |

elementary abelian group:E8 | 5 | 1 | 1 | 3 | 1/16 |

## Presentations

`Further information: presentations for groups of order 8`

Below are the power-commutator presentations for groups of order 8.

Group | Second part of GAP ID (GAP ID is (p^3,2nd part) | Nilpotency class | Minimum size of generating set | Prime-base logarithm of exponent | full power-commutator presentation | ||||
---|---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 1 | 1 | 3 | 1 | 0 | 1 | 0 | [SHOW MORE] |

direct product of Z4 and Z2 | 2 | 1 | 2 | 2 | 0 | 1 | 0 | 0 | [SHOW MORE] |

dihedral group:D8 | 3 | 2 | 2 | 2 | 0 | 0 | 0 | 1 | [SHOW MORE] |

quaternion group | 4 | 2 | 2 | 2 | 0 | 1 | 1 | 1 | [SHOW MORE] |

elementary abelian group:E8 | 5 | 1 | 3 | 1 | 0 | 0 | 0 | 0 | [SHOW MORE] |

## Subgroup/quotient relationships

### Subgroup relationships

### Quotient relationships

## Arithmetic functions

### Functions taking values between 0 and 3

Group | GAP ID (second part) | Hall-senior number | prime-base logarithm of exponent | nilpotency class | derived length | Frattini length | minimum size of generating set | subgroup rank | rank as p-group | normal rank | characteristic rank | prime-base logarithm of order of derived subgroup | prime-base logarithm of order of inner automorphism group |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Cyclic group:Z8 | 1 | 3 | 3 | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 0 | 0 |

Direct product of Z4 and Z2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 |

Dihedral group:D8 | 3 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 |

Quaternion group | 4 | 5 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 |

Elementary abelian group:E8 | 5 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 0 | 0 |

Mean (with equal weight on all groups) | 2 | 1.4 | 1.4 | 2 | 2 | 2 | 1.8 | 1.8 | 1.6 | 0.4 | 0.8 | ||

Mean (using cohomology tree probability distribution) | 2.1875 | 1.25 | 1.25 | 2.1875 | 1.8125 | 1.8125 | 1.75 | 1.75 | 1.5625 | 0.25 | 0.5 |

Same, with rows and columns interchanged:

Function | Cyclic group:Z8 | Direct product of Z4 and Z2 | Dihedral group:D8 | Quaternion group | Elementary abelian group:E8 |
---|---|---|---|---|---|

prime-base logarithm of exponent | 3 | 2 | 2 | 2 | 1 |

nilpotency class | 1 | 1 | 2 | 2 | 1 |

derived length | 1 | 1 | 2 | 2 | 1 |

Frattini length | 3 | 2 | 2 | 2 | 1 |

minimum size of generating set | 1 | 2 | 2 | 2 | 3 |

subgroup rank | 1 | 2 | 2 | 2 | 3 |

rank as p-group | 1 | 2 | 2 | 1 | 3 |

normal rank as p-group | 1 | 2 | 2 | 1 | 3 |

characteristic rank as p-group | 1 | 2 | 1 | 1 | 3 |

### Arithmetic function values of a counting nature

Group | GAP ID (second part) | Hall-senior number | number of conjugacy classes | number of subgroups | number of conjugacy classes of subgroups | number of normal subgroups | number of automorphism classes of subgroups | number of characteristic subgroups |
---|---|---|---|---|---|---|---|---|

Cyclic group:Z8 | 1 | 3 | 8 | 4 | 4 | 4 | 4 | 4 |

Direct product of Z4 and Z2 | 2 | 2 | 8 | 8 | 8 | 8 | 6 | 4 |

Dihedral group:D8 | 3 | 4 | 5 | 10 | 8 | 6 | 6 | 4 |

Quaternion group | 4 | 5 | 5 | 6 | 6 | 6 | 4 | 3 |

Elementary abelian group:E8 | 5 | 1 | 8 | 16 | 16 | 16 | 4 | 2 |

Here is the same table, with rows and columns interchanged:

Function | Cyclic group:Z8 | Direct product of Z4 and Z2 | Dihedral group:D8 | Quaternion group | Elementary abelian group:E8 |
---|---|---|---|---|---|

number of conjugacy classes | 8 | 8 | 5 | 5 | 8 |

number of subgroups | 4 | 8 | 10 | 6 | 16 |

number of conjugacy classes of subgroups | 4 | 8 | 8 | 6 | 16 |

number of normal subgroups | 4 | 8 | 6 | 6 | 16 |

number of automorphism classes of subgroups | 4 | 6 | 6 | 4 | 4 |

number of characteristic subgroups | 4 | 4 | 4 | 3 | 2 |

### Arithmetic function values of a representational nature

Group | GAP ID (second part) | Hall-senior number | minimum degree of faithful permutation representation | minimum degree of faithful transitive permutation representation | minimum degree of faithful linear representation over | symmetric genus |
---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 8 | 8 | 1 | ? |

direct product of Z4 and Z2 | 2 | 2 | 6 | 8 (at most) | 2 | ? |

dihedral group:D8 | 3 | 4 | 4 | 4 | 2 | ? |

quaternion group | 4 | 5 | 8 | 8 | 2 | ? |

elementary abelian group:E8 | 5 | 1 | 6 | 8 (at most) | 3 | ? |

### Numerical invariants

Group | Conjugacy class sizes | Degrees of irreducible representations |
---|---|---|

cyclic group:Z8 | 1 (8 times) | 1 (8 times) |

direct product of Z4 and Z2 | 1 (8 times) | 1 (8 times) |

dihedral group:D8 | 1,1,2,2,2 | 1,1,1,1,2 |

quaternion group | 1,1,2,2,2 | 1,1,1,1,2 |

elementary abelian group:E8 | 1 (8 times) | 1 (8 times) |

## Group properties

Property | Cyclic group:Z8 | Direct product of Z4 and Z2 | Dihedral group:D8 | Quaternion group | Elementary abelian group:E8 |
---|---|---|---|---|---|

cyclic group | Yes | No | No | No | No |

elementary abelian group | No | No | No | No | Yes |

abelian group | Yes | Yes | No | No | Yes |

homocyclic group | Yes | No | No | No | Yes |

metacyclic group | Yes | Yes | Yes | Yes | No |

metabelian group | Yes | Yes | Yes | Yes | Yes |

group of nilpotency class two | Yes | Yes | Yes | Yes | Yes |

maximal class group | No | No | Yes | Yes | No |

ambivalent group | No | No | Yes | Yes | Yes |

rational group | No | No | Yes | Yes | Yes |

rational-representation group | No | No | Yes | No | Yes |

group in which every element is automorphic to its inverse | Yes | Yes | Yes | Yes | Yes |

group in which any two elements generating the same cyclic subgroup are automorphic | Yes | Yes | Yes | Yes | Yes |

T-group | Yes | Yes | No | Yes | Yes |

C-group | No | No | No | No | Yes |

SC-group | No | No | No | No | Yes |

UL-equivalent group | Yes | Yes | Yes | Yes | Yes |

algebra group | No | Yes | Yes | No | Yes |

## Families and classification

`Further information: Classification of groups of order 8`

### Up to isoclinism

FACTS TO CHECK AGAINSTfor isoclinic groups (groups with an isoclinism between them):

by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.

isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations

FACTS TO CHECK AGAINSTfor isoclinic groups (groups with an isoclinism between them):

by definition, isoclinic groups have isomorphic inner automorphism groups and isomorphic derived subgroups, with the isomorphisms compatible.

isoclinic groups have same nilpotency class|isoclinic groups have same derived length | isoclinic groups have same proportions of conjugacy class sizes | isoclinic groups have same proportions of degrees of irreducible representations

The equivalence classes up to being isoclinic were classified by Hall and Senior, and we call them Hall-Senior families.

Family name | Isomorphism class of inner automorphism group | Isomorphism class of derived subgroup | Number of groups | Nilpotency class | Members | Second part of GAP ID of members (sorted ascending) | Hall-Senior numbers of members (sorted ascending) | Smallest order of group isoclinic to these groups | Stem groups (groups of smallest order) |
---|---|---|---|---|---|---|---|---|---|

(abelian groups) | trivial group | trivial group | 3 | 1 | cyclic group:Z8, direct product of Z4 and Z2, elementary abelian group:E8 | 1,2,5 | 1-3 | 1 | trivial group |

Klein four-group | cyclic group:Z2 | 2 | 2 | dihedral group:D8, quaternion group | 3,4 | 4,5 | 8 | dihedral group:D8, quaternion group | |

Total (2 rows) | -- | -- | 5 | -- | -- | -- | -- | -- | -- |

### Up to Hall-Senior genus

Up to the relation of groups having the same Hall-Senior genus, there are four equivalence classes:

Genus name | Description of genus | Members | Hall-Senior numbers | Second parts of GAP ID of members |
---|---|---|---|---|

cyclic group | cyclic group:Z8 | 3 | 1 | |

abelian group for partition | direct product of Z4 and Z2 | 2 | 2 | |

(the dihedral group is and the quaternion group is ) | non-abelian groups | dihedral group:D8, quaternion group | 4,5 | 3,4 |

elementary abelian group | elementary abelian group:E8 | 1 | 5 |

### Up to isologism for higher class

Since all the groups of order 8 has class at most two, we have a unique equivalence class under isologism for any class equal to or more than two.

### Up to isologism for elementary abelian groups

Each of the abelian groups is in a different equivalence class under the equivalence relation of being isologic with respect to elementary abelian 2-groups. The two non-abelian groups are isologic to each other with respect to the variety of elementary abelian 2-groups.

### Cohomology tree

## Element structure

`Further information: element structure of groups of order 8`

### Order statistics

FACTS TO CHECK AGAINST:

ORDER STATISTICS(cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots

1-ISOMORPHISM(cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic

Here are the statistics for a *particular* order.

Group | Second part of GAP ID | Hall-Senior number | Number of elements of order 1 | Number of elements of order 2 | Number of elements of order 4 | Number of elements of order 8 |
---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 1 | 2 | 4 |

direct product of Z4 and Z2 | 2 | 2 | 1 | 3 | 4 | 0 |

dihedral group:D8 | 3 | 4 | 1 | 5 | 2 | 0 |

quaternion group | 4 | 5 | 1 | 1 | 6 | 0 |

elementary abelian group:E8 | 5 | 1 | 1 | 7 | 0 | 0 |

Here are the *number of root* statistics. The number of roots equals the number of elements whose order divides .

Group | Second part of GAP ID | Hall-Senior number | Number of first roots | Number of roots | Number of roots | Number of roots |
---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 2 | 4 | 8 |

direct product of Z4 and Z2 | 2 | 2 | 1 | 4 | 8 | 8 |

dihedral group:D8 | 3 | 4 | 1 | 6 | 8 | 8 |

quaternion group | 4 | 5 | 1 | 2 | 8 | 8 |

elementary abelian group:E8 | 5 | 1 | 1 | 8 | 8 | 8 |

### Equivalence classes

No two of the groups of order 8 are order statistics-equivalent, and hence no two of them are 1-isomorphic.

## Subgroup structure

`Further information: subgroup structure of groups of order 8`

## Linear representation theory

`Further information: linear representation theory of groups of order 8`

Group | GAP ID second part | Hall-Senior number | Nilpotency class | Degrees as list | Number of irreps of degree 1 (= order of abelianization) | Number of irreps of degree 2 | Total number of irreps (= number of conjugacy classes) |
---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 3 | 1 | 1,1,1,1,1,1,1,1 | 8 | 0 | 8 |

direct product of Z4 and Z2 | 2 | 2 | 1 | 1,1,1,1,1,1,1,1 | 8 | 0 | 8 |

dihedral group:D8 | 3 | 4 | 2 | 1,1,1,1,2 | 4 | 1 | 5 |

quaternion group | 4 | 5 | 2 | 1,1,1,1,2 | 4 | 1 | 5 |

elementary abelian group:E8 | 5 | 1 | 1 | 1,1,1,1,1,1,1,1 | 8 | 0 | 8 |

## Subgroup-defining functions

### Values up to isomorphism type

## Automorphism groups

Group | GAP ID (second part) | Order of automorphism group | Iso. class of automorphism group | Log_2 of largest power of 2 dividing automorphism group | Iso. class of 2-Sylow subgroup of automorphism group | Log_2 of order of 2-core | Iso. class of 2-core | Log_2 of order of inner automorphism group | Iso. class of inner automorphism group |
---|---|---|---|---|---|---|---|---|---|

cyclic group:Z8 | 1 | 4 | Klein four-group | 2 | Klein four-group | 2 | Klein four-group | 0 | trivial group |

direct product of Z4 and Z2 | 2 | 8 | dihedral group:D8 | 3 | dihedral group:D8 | 3 | dihedral group:D8 | 0 | trivial group |

dihedral group:D8 | 3 | 8 | dihedral group:D8 | 3 | dihedral group:D8 | 3 | dihedral group:D8 | 2 | Klein four-group |

quaternion group | 4 | 24 | symmetric group:S4 | 3 | dihedral group:D8 | 2 | Klein four-group | 2 | Klein four-group |

elementary abelian group:E8 | 5 | 168 | general linear group:GL(3,2) | 3 | dihedral group:D8 | 0 | trivial group | 0 | trivial group |