Finite Group DatabasebetaFinite Group Databasebeta| order | 4032 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×ΓL(1,𝔽8)×GL(2,ℤ/4) |
| Fitting | C25×C7 ⋊ C3×D3 |
| Sylow | C7 ⋊ C2×C3×GL(2,ℤ/4) |
1148 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 105 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) = 168 |
A minimal faithful permutation representation — 3 generators on 17 points:
(4,6)(7,9,10)(12,13,14)(15,17,16);(3,4)(5,6)(8,9)(11,12,14,16,13,15,17);(1,2)140 conjugacy classes — view the full character table.
| maximal subgroup of | C2×ΓL(1,𝔽8)×A4⋊S4, C2×AGL(1,𝔽7)×GL(2,ℤ/4) |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×GL(2,ℤ/4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,6)(7,9,10)(12,13,14)(15,17,16), (3,4)(5,6)(8,9)(11,12,14,16,13,15,17), (1,2));