Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of #226 |
| automorphism | Aut(G) = C22×S4 (order 96) · Inn(G) = S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #41 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22 |
| Frattini, socle | C2 | C2×S4 |
| Fitting | #41 | D3 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C4 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,6,12,8,3,14,5,15,10,2,11,16)(4,7,9,13);(3,4)(6,7)(9,11)(10,12)(13,15)(14,16)Polycyclic presentation:
2^2,3,2^3|b^2=f,d^2=f,e^2=f,c^a=c^2,d^a=e,e^a=d,d^c=ef,e^c=de,e^d=ef16 conjugacy classes — view the full character table.
| maximal subgroup of | #5404, #5405, C2×#985, … |
| maximal normal subgroup of | #5404, #5405, C2×#985, … |
| maximal quotient of | C4×#226, #5389, C4×GL(2,𝔽3), … |
| outer automorphism group of | C3⋊D15 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 192);