Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D4×S4 (order 192) · Inn(G) = C2×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22 |
| Fitting | #41 | D3 |
| prime radical | #41⋊C3 | C2 |
| hypercenter | C4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,12,18,14,4,28,6,29,16,3,17,30)(2,19,24,11,8,31,10,25,22,7,23,32)(5,13,15,27)(9,20,21,26);(1,2,6,10)(3,11,14,7)(4,9,17,21)(5,8,15,23)(12,26,29,20)(13,25,27,19)(16,24,18,22)(28,31,30,32)Polycyclic presentation:
2^2,3,2^3|a^2=f,b^2=f,d^2=f,e^2=f,b^a=bf,c^a=c^2,d^a=e,e^a=d,d^c=ef,e^c=de,e^d=ef13 conjugacy classes — view the full character table.
| maximal subgroup of | #5403, #5406, #5428, … |
| maximal normal subgroup of | #5403, #5406, #5428, … |
| maximal quotient of | #5386, #5388, C4⋊#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 191);