Finite Group DatabasebetaFinite Group Databasebeta| order | 32 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#120 (order 64) · Inn(G) = C22 · Out(G) = C2×D4 |
| holomorph | G ⋊ Aut(G) = #72219555 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C22 |
| commutator | C2 | C2×C8 |
| socle, prime radical | C22 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
19 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,4)(5,6,7,9,8,10,11,12);(1,2,3,4)Polycyclic presentation:
2^5|a^2=d,b^2=c,d^2=e,b^a=bc20 conjugacy classes — view the full character table.
| Fitting subgroup of | C4⋊C8×A5, C4⋊#91145, #14514991, … |
| maximal subgroup of | D4⋊C8, Q8⋊C8, #331, … |
| maximal normal subgroup of | D4⋊C8, Q8⋊C8, #331, … |
| maximal quotient of | #334, #335, #336, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(32, 12);