Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C29 |
| direct product | C23×#5948 |
| Fitting, Sylow | C25×#41 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C24×C3 |
| Frattini | C2 | C24×C22⋊A4 |
| Fitting | C25×#41 | C3 |
| socle | C26 | C2×A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C23×C4 | |
152803 conjugacy classes of subgroups; only the 638 normal subgroups are shown.
This group has 152803 conjugacy classes of subgroups — too many to draw. Only its 638 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) = 192 |
A minimal faithful permutation representation — 4 generators on 26 points:
(5,6)(8,10,9)(13,14,21)(16,17,25)(19,20,22)(23,24,26);(3,4)(11,19,15,21)(12,23,18,25)(13,22,20,14)(16,26,24,17);(1,2)(7,10)(8,9);(1,2)(3,4)(5,6)(11,12,15,18)(13,16,20,24)(14,17,22,26)(19,23,21,25)192 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#87545, C2×#72227572, #72267948 |
| maximal normal subgroup of | C23×#87545, C2×#72227572, #72267948 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(8,10,9)(13,14,21)(16,17,25)(19,20,22)(23,24,26), (3,4)(11,19,15,21)(12,23,18,25)(13,22,20,14)(16,26,24,17), (1,2)(7,10)(8,9), (1,2)(3,4)(5,6)(11,12,15,18)(13,16,20,24)(14,17,22,26)(19,23,21,25));