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 $$ab$$.

Subgroups

 * 1) The subgroup::trivial group. (1)
 * 2) The subgroup $$\langle b \rangle$$, which is the unique characteristic subgroup of order $$2$$. Isomorphic to subgroup::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 quotient group::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 subgroup::cyclic group:Z2. The quotient group for each is isomorphic to quotient group::elementary abelian group:E8. (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 subgroup::cyclic group:Z2. (4)
 * 5) The subgroup $$\langle a^2,b \rangle$$, which is the center, first agemo subgroup, and Frattini subgroup. Isomorphic to subgroup::Klein four-group. The quotient group is isomorphic to quotient group::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 subgroup::Klein four-group. The quotient group is isomorphic to quotient group::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 subgroup::Klein four-group. The quotient group is also isomorphic to a quotient group::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 subgroup::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 subgroup::direct product of Z4 and Z2. The quotient group is isomorphic to quotient group::cyclic group:Z2. (2)
 * 10) The subgroup $$\langle a^2,b,c \rangle$$. Isomorphic to subgroup::elementary abelian group:E8. The quotient group is isomorphic to quotient group::cyclic group:Z2. (1)
 * 11) The whole group. (1)

Other descriptions
The group can be constructed using the following GAP commands:

gap> F := FreeGroup(3);  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)]; 