Finite Group DatabasebetaFinite Group Databasebeta| order | 12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23×C3 |
| direct product | C2×ΓL(1,𝔽8)×A4⋊S4 |
| Fitting, socle | C25×C7 ⋊ C3×C3⋊D3 |
| derived | C7×A42 ⋊ C22×C3 |
| Sylow | C7 ⋊ C2×C3×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | ΓL(1,𝔽8)×A4⋊S4 |
| commutator, cosocle, hypocenter | C7×A42 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C25×C7 | C3×C3⋊D3 |
| LCS | G ▷ C7×A42 | |
| UCS | C1 ◁ C2 | |
1516 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1516 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,3,4)(5,7,6)(10,13,12,11)(14,17,15,16);(1,2,4,6,3,5,7)(8,9)(15,16,17);(11,12,13)(15,16,17)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×AGL(1,𝔽7)×A4⋊S4 |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×A4⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3,4)(5,7,6)(10,13,12,11)(14,17,15,16), (1,2,4,6,3,5,7)(8,9)(15,16,17), (11,12,13)(15,16,17));