Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C22×D3×D4 (order 192) · Inn(G) = C2×D3 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×C8 ⋊ C2 |
| Sylow | C3 ⋊ #131 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C2×D3 |
| commutator, socle | C2×C3 | C22×C4 |
| Frattini | C4 | C22×D3 |
| Fitting | C2×C3×C8 | C2 |
| prime radical | #235 | C2 |
| cosocle | C3×C4 | C23 |
| hypercenter | C2×C8 | D3 |
| hypocenter | C3 | #131 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C8 ◁ C2×C8 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,6,7,12,8,13,14,18)(5,10,11,17,9,15,16,19);(5,9)(10,15)(11,16)(17,19);(2,3)(4,5)(6,10)(7,11)(8,9)(12,17)(13,15)(14,16)(18,19)Polycyclic presentation:
2^5,3|c^2=d,d^2=e,b^a=be,f^a=f^236 conjugacy classes — view the full character table.
| maximal subgroup of | #4508, #4515, #4700, … |
| maximal normal subgroup of | #4508, #4515, #4700, … |
| maximal quotient of | C8×Dic6, #4679, C8×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 108);