Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = #23900 (order 256) · Inn(G) = C22 · Out(G) = D42 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22 |
| commutator | C2 | C22×C4 |
| Frattini, socle, cosocle | C22 | C23 |
| prime radical | C23 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^5|a^2=e,c^2=d,b^a=bd20 conjugacy classes — view the full character table.
| derived subgroup of | #94449, #94450, #94451, … |
| Fitting subgroup of | #117×A5, #14515087, #14515263 |
| maximal subgroup of | H(3,ℤ/4), #339, #344, … |
| maximal normal subgroup of | H(3,ℤ/4), #339, #344, … |
| maximal quotient of | #376, C4×C22⋊C4, C4×C4⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 24);