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SmallGroup(16,3)

Definition

This group can be defined using the following presentation:

G := \langle a,b,c \mid a^4 = b^2 = c^2 = e, ab = ba, bc = cb, cac^{-1} = ab \rangle

Note that G is generated by a,c alone, because the final relation allows us to write b in terms of a and c.

The subgroup \langle a,b \rangle is isomorphic to the direct product of Z4 and Z2, and the element c is an element of order two that acts on the subgroup \langle a,b \rangle by conjugation by fixing b and sending a to ab.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 16#Arithmetic functions
Function Value Similar groups Explanation for function value
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 16 groups with same order
prime-base logarithm of order 4 groups with same prime-base logarithm of order
max-length of a group 4 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 4 chief length equals prime-base logarithm of order for group of prime power order
composition length 4 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 2 groups with same order and prime-base logarithm of exponent | groups with same prime-base logarithm of order and prime-base logarithm of exponent | groups with same prime-base logarithm of exponent
nilpotency class 2 groups with same order and nilpotency class | groups with same prime-base logarithm of order and nilpotency class | groups with same nilpotency class
derived length 2 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length 2 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set
subgroup rank of a group 3 groups with same order and subgroup rank of a group | groups with same prime-base logarithm of order and subgroup rank of a group | groups with same subgroup rank of a group
rank of a p-group 3 groups with same order and rank of a p-group | groups with same prime-base logarithm of order and rank of a p-group | groups with same rank of a p-group
normal rank of a p-group 3 groups with same order and normal rank of a p-group | groups with same prime-base logarithm of order and normal rank of a p-group | groups with same normal rank of a p-group
characteristic rank of a p-group 3 groups with same order and characteristic rank of a p-group | groups with same prime-base logarithm of order and characteristic rank of a p-group | groups with same characteristic rank of a p-group
number of conjugacy classes 10 groups with same order and number of conjugacy classes | groups with same prime-base logarithm of order and number of conjugacy classes | groups with same number of conjugacy classes
number of conjugacy classes of subgroups 17 groups with same order and number of conjugacy classes of subgroups | groups with same prime-base logarithm of order and number of conjugacy classes of subgroups | groups with same number of conjugacy classes of subgroups

Group properties

Subgroups

Further information: subgroup structure of SmallGroup(16,3)

  1. The trivial group. (1)
  2. The subgroup \langle b \rangle, which is the unique characteristic subgroup of order 2. Isomorphic to cyclic group:Z2. It is the commutator subgroup, and can also be described as the unique group of order two containing an element that is not a square but is a product of squares. The quotient group is isomorphic to direct product of Z4 and Z2. (1)
  3. The subgroups \langle a^2 \rangle and \langle a^2b \rangle, which are both normal subgroups related by an outer automorphism. Isomorphic to cyclic group:Z2. The quotient group for each is isomorphic to dihedral group:D8. (2)
  4. The subgroups \langle c \rangle, \langle bc \rangle, \langle a^2c \rangle and \langle a^2bc \rangle. Neither is normal, and they come in two conjugacy classes of size two each. Isomorphic to cyclic group:Z2. (4)
  5. The subgroup \langle a^2,b \rangle, which is the center, first agemo subgroup, and Frattini subgroup. Isomorphic to Klein four-group. The quotient group is isomorphic to Klein four-group. (1)
  6. The subgroups \langle b,c \rangle and \langle b, a^2c\rangle. Both are normal subgroups and are related by an outer automorphism. Isomorphic to Klein four-group. The quotient group is isomorphic to cyclic group:Z4. (2)
  7. The subgroups \langle a^2,c \rangle, \langle a^2,bc \rangle, \langle a^2b,c\rangle, and \langle a^2b, bc \rangle. Two conjugacy classes of size two each. Isomorphic to Klein four-group. The quotient group is also isomorphic to a Klein four-group. (4)
  8. The subgroups \langle a \rangle, \langle ab \rangle, \langle ac \rangle, and \langle abc \rangle. All of them are related by outer automorphisms, and they form two conjugacy classes of subgroups of size two each: \langle a \rangle is conjugate to \langle ab \rangle, while \langle ac \rangle is conjugate to \langle abc \rangle. Isomorphic to cyclic group:Z4. (4)
  9. The subgroups \langle a,b \rangle and \langle ac, b \rangle. These are both normal subgroups related by an outer automorphism. Isomorphic to direct product of Z4 and Z2. The quotient group is isomorphic to cyclic group:Z2. (2)
  10. The subgroup \langle a^2,b,c \rangle. Isomorphic to elementary abelian group:E8. The quotient group is isomorphic to cyclic group:Z2. (1)
  11. The whole group. (1)

Subgroup-defining functions

GAP implementation

Group ID

This finite group has order 16 and has ID 3 among the groups of order 16 in GAP's SmallGroup library. For context, there are 14 groups of order 16. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(16,3)

For instance, we can use the following assignment in GAP to create the group and name it G:

gap> G := SmallGroup(16,3);

Conversely, to check whether a given group G is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [16,3]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.


Other descriptions

The group can be constructed using the following GAP commands:

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4, F.2^2, F.1*F.2*F.1^(-1)*F.2^(-1),F.3^2,F.3*F.2*F.3^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.2^(-1)*F.1^(-1)];
<fp group on the generators [ f1, f2, f3 ]>