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) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived | SL(2,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | S4 |
| commutator, hypocenter | SL(2,𝔽3) | C4 |
| Fitting | C2×Q8 | D3 |
| prime radical, cosocle | C2×SL(2,𝔽3) | C2 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,3,4)(5,6,12,7,8,11,10,9);(6,9,12)(7,10,11)Polycyclic presentation:
2^2,3,2^3|a^2=b,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 | #5386, C4×#226, #5388, … |
| maximal normal subgroup of | #5386, C4×#226, #5388, … |
| maximal quotient of | SL(2,𝔽3)⋊C8, C2×#859, #63421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 66);