Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
| direct product | C23×#5935 |
| Fitting | C23×H(5,𝔽2) ⋊ D3 |
| derived | #997 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | #1020 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C2 | C23×#1020 |
| Fitting | C23×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C24 | |
16193 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 16193 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 14 points:
(5,6)(7,9,10)(8,13,12);(3,4)(7,13,8,9)(10,14,12,11);(9,12)(10,13)(11,14);(1,2)(7,8)(9,13)(10,12)(11,14)104 conjugacy classes — view the full character table.
| maximal subgroup of | C23×SO+(4,𝔽3), #72258828, C22×#11715828, … |
| maximal normal subgroup of | #72258828, C22×#11715828, C22×#11714182, … |
| maximal quotient of | C2×D4×#5935 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,9,10)(8,13,12), (3,4)(7,13,8,9)(10,14,12,11), (9,12)(10,13)(11,14), (1,2)(7,8)(9,13)(10,12)(11,14));