Groupprops, The Group Properties Wiki (pre-alpha)
YOUR FEEDBACK IS IMPORTANT!
Please take a short user satisfaction survey about Groupprops.
Your survey responses will be helpful in improving the site experience!
Thanks in advance!
Groups of order 8
From Groupprops
This article gives basic information comparing and contrasting the groups of order 8.
Contents |
The list
| Common name for group | GAP ID |
|---|---|
| cyclic group:Z8 | (8,1) |
| direct product of Z4 and Z2 | (8,2) |
| dihedral group:D8 | (8,3) |
| quaternion group | (8,4) |
| elementary abelian group:E8 | (8,5) |
Arithmetic functions
| Function | Cyclic group:Z8 | Direct product of Z4 and Z2 | Dihedral group:D8 | Quaternion group | Elementary abelian group:E8 |
|---|---|---|---|---|---|
| order | 8 | 8 | 8 | 8 | 8 |
| exponent | 8 | 4 | 4 | 4 | 2 |
| derived length | 1 | 1 | 2 | 2 | 1 |
| nilpotency class | 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 |
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 |
Possibilities for maximal subgroups
| Collection of isomorphism classes of maximal subgroups | Groups |
|---|---|
| cyclic group:Z4 only | cyclic group:Z8, quaternion group |
| Klein four-group only | elementary abelian group:E8 |
| cyclic group:Z4 and Klein four-group | direct product of Z4 and Z2, dihedral group:D8 |
