Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×D3×#120 (order 384) · Inn(G) = C2×D3 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C3 ⋊ #97 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2×C4 | C2×D3 |
| commutator | C2×C3 | C42 |
| Fitting | C2×C3×C8 | C2 |
| socle, prime radical | C22×C3 | C2×C4 |
| cosocle | C2×C3×C4 | C22 |
| hypercenter | C2×C8 | D3 |
| hypocenter | C3 | #97 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C8 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,3)(8,9,10,13,11,12,14,15);(2,3)(4,5,6,7)(9,12)(13,15)Polycyclic presentation:
2^5,3|a^2=c,b^2=d,d^2=e,b^a=be,f^a=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | #4512, #4679, C4×ΓL(1,𝔽25), … |
| maximal normal subgroup of | #4512, #4679, C4×ΓL(1,𝔽25), … |
| maximal quotient of | #4454, #4455, #4512, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 22);