Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11711579 |
| derived | #997 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23 | #1020⋊C2 |
| commutator, hypocenter | #997 | C22×C4 |
| Frattini | C24 | C22×S4 |
| Fitting | C22×B(4,𝔽2) | D3 |
| prime radical | C22×B(4,𝔽2)⋊C3 | C2 |
| cosocle | C2×#997 | C23 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C23 | |
2242 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 2242 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) = 14 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 14 points:
(3,4,5,6)(7,11,13,9,10,8,14,12);(8,12,14,11,9,13);(1,2)(8,11)(9,12)(13,14)80 conjugacy classes — view the full character table.
| maximal subgroup of | #72341407, C2×C4×#73026, #72341428, … |
| maximal normal subgroup of | #72341407, C2×C4×#73026, #72341428, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4,5,6)(7,11,13,9,10,8,14,12), (8,12,14,11,9,13), (1,2)(8,11)(9,12)(13,14));