Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×#23467⋊C3 |
| Fitting, Sylow | C2×#23467 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #120⋊C3 |
| commutator, cosocle, hypocenter | #543 | C23×C3 |
| Frattini | C24 | C23×A4 |
| Fitting | C2×#23467 | C3 |
| socle | C23 | C2×#120⋊C3 |
| prime radical | C2×#2029⋊C3 | C2 |
| LCS | G ▷ #543 | |
| UCS | C1 ◁ C22×C4 | |
1998 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1998 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2)(4,7,5)(8,9,10)(13,20,14)(16,23,17)(19,25,21)(22,26,24);(3,9,6,7)(4,5,8,10)(11,23,15,26)(12,25,18,20)(13,17,19,24)(14,16,21,22);(13,19)(14,21)(16,22)(17,24)(20,25)(23,26)120 conjugacy classes — view the full character table.
| maximal subgroup of | #72268645 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2)(4,7,5)(8,9,10)(13,20,14)(16,23,17)(19,25,21)(22,26,24), (3,9,6,7)(4,5,8,10)(11,23,15,26)(12,25,18,20)(13,17,19,24)(14,16,21,22), (13,19)(14,21)(16,22)(17,24)(20,25)(23,26));