Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C22×#10619528 |
| derived, Sylow | C92 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D92 |
| commutator, hypocenter | C92 | C23×C4 |
| Frattini | C2×C32 | C22×D32 |
| Fitting | C23×C92 | C22 |
| socle | C23×C32 | D32 |
| prime radical | C23×C9⋊D9 | C2 |
| cosocle | C2×C92 | C24 |
| LCS | G ▷ C92 | |
| UCS | C1 ◁ C23 | |
1531 conjugacy classes of subgroups; only the 290 normal subgroups are shown.
This group has 1531 conjugacy classes of subgroups — too many to draw. Only its 290 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) = 144 |
A minimal faithful permutation representation — 4 generators on 26 points:
(10,12)(11,15)(13,16)(14,17)(18,24)(19,21)(20,26)(22,25);(3,4)(9,14,12,11,16,13,15,10,17);(1,2);(5,6,7,8)(19,20)(21,22)(23,25)(24,26)288 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×#72219961 |
| maximal normal subgroup of | C2×#72219961 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((10,12)(11,15)(13,16)(14,17)(18,24)(19,21)(20,26)(22,25), (3,4)(9,14,12,11,16,13,15,10,17), (1,2), (5,6,7,8)(19,20)(21,22)(23,25)(24,26));