Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×#73026 (order 2304) · Inn(G) = C23×D3 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C2×C3×Q8 ⋊ C2 |
| Sylow | C3 ⋊ 2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D3 |
| commutator, socle, cosocle | C2×C3 | C24 |
| Fitting | C2×C3×Q8 | C2 |
| hypercenter | C2×Q8 | D3 |
| hypocenter | C3 | 2−1+4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
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)(5,11)(9,17)(10,19)(16,18);(4,7,8,15)(5,10,11,19)(6,12,13,14)(9,16,17,18);(4,6,8,13)(5,9,11,17)(7,14,15,12)(10,18,19,16);(2,3)(4,5)(6,9)(7,10)(8,11)(12,16)(13,17)(14,18)(15,19)Polycyclic presentation:
2^5,3|b^2=e,c^2=e,d^a=de,f^a=f^2,c^b=ce27 conjugacy classes — view the full character table.
| maximal subgroup of | #4752, #4753, #5062, … |
| maximal normal subgroup of | #4752, #4753, #5062, … |
| maximal quotient of | #5500, #5504, #5507, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 214);