Finite Group DatabasebetaFinite Group Databasebeta| order | 12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C23×C33 |
| direct product | C2×A42×AGL(1,𝔽7) |
| Fitting, socle | C25×C7 ⋊ C2×C33 |
| derived | C24×C7 ⋊ C22×C33 |
| Hall | C25×D7 ⋊ C33 |
| Sylow | C7 ⋊ C22×C3×A42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A42×AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C24×C7 | C22×C33 |
| Frattini | C1 | G |
| Fitting, socle | C25×C7 | C2×C33 |
| LCS | G ▷ C24×C7 | |
| UCS | C1 ◁ C2 | |
3284 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 3284 conjugacy classes of subgroups — too many to draw. Only its 232 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:
(1,2)(3,5)(6,7)(10,13)(11,12)(14,17,15);(8,9)(11,13,12)(15,17,16);(2,4,5)(3,6,7)224 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((1,2)(3,5)(6,7)(10,13)(11,12)(14,17,15), (8,9)(11,13,12)(15,17,16), (2,4,5)(3,6,7));