Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of #1020 |
| automorphism | Aut(G) = (#1020⋊C3)⋊C2 (order 576) · Inn(G) = G · Out(G) = D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, socle | C24 ⋊ D3 |
| derived | C22⋊A4 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22⋊A4 | C2 |
| Fitting, socle | C24 | D3 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 8 |
A minimal faithful permutation representation — 2 generators on 8 points:
(1,7,3,2)(4,8,5,6);(3,4,5)(6,7,8)Polycyclic presentation:
2,3,2^4|b^a=b^2,c^a=d,d^a=c,e^a=f,f^a=e,c^b=d,d^b=cd,e^b=ef,f^b=e| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9 | C10 | |
|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 3 | 3 | 6 | 12 | 12 | 12 | 12 | 32 |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 3 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 |
| χ3 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 |
| χ4 | 3 | 3 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 0 |
| χ5 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 0 |
| χ6 | 3 | -1 | 3 | -1 | -1 | 1 | -1 | 1 | -1 | 0 |
| χ7 | 3 | -1 | 3 | -1 | -1 | -1 | 1 | -1 | 1 | 0 |
| χ8 | 3 | -1 | -1 | 3 | -1 | 1 | -1 | -1 | 1 | 0 |
| χ9 | 3 | -1 | -1 | 3 | -1 | -1 | 1 | 1 | -1 | 0 |
| χ10 | 6 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 |
| maximal subgroup of | #1020⋊C2, C2×#1020, #1020⋊C3, … |
| maximal normal subgroup of | #1020⋊C2, C2×#1020, #1020⋊C3, … |
| maximal quotient of | #5930, #5931, #5932, … |
| outer automorphism group of | #560 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 227);