Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×S4 (order 96) · Inn(G) = C2×S4 · Out(G) = C2 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C2×Q8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C2 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C22 |
| Fitting | C2×Q8 | D3 |
| hypercenter | C22 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
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:
(1,3,10)(2,13,15,6,7,16)(5,11,12)(8,14);(1,2)(3,8)(4,7)(5,6)(9,13)(10,16)(11,14)(12,15)Polycyclic presentation:
2^2,3,2^3|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 | C2×#983, #5923, #5926, … |
| maximal normal subgroup of | C2×#983, #5923, #5926, … |
| maximal quotient of | #5388, #5389, #5394, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 190);