Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = C23×D3 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D3 |
| commutator, socle, cosocle | C2×C3 | C24 |
| Fitting | C3×#41 | C2 |
| hypercenter | #41 | D3 |
| hypocenter | C3 | 2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 | |
146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(4,8,9,7)(5,11,12,16)(6,13,14,15)(10,18,19,17);(4,7,9,8)(5,11,12,16)(6,13,14,15)(10,17,19,18);(4,6,9,14)(5,10,12,19)(7,15,8,13)(11,18,16,17);(2,3)(4,5)(6,10)(7,11)(8,16)(9,12)(13,17)(14,19)(15,18)Polycyclic presentation:
2^5,3|b^2=e,c^2=e,d^2=e,d^a=de,f^a=f^2,c^b=ce27 conjugacy classes — view the full character table.
| maximal subgroup of | #4826, #4827, #5201, … |
| maximal normal subgroup of | #4826, #4827, #5201, … |
| maximal quotient of | #5516, #5518, #5519, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 217);