Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C27 |
| direct product | C22×#2380⋊C3 |
| Fitting, Sylow | C22×#2380 ⋊ C3 |
| derived | C42 ⋊ C23×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #560⋊C3 |
| commutator, hypocenter | C42 | C23×C3×C4 |
| Frattini | C23 | C24×A4 |
| Fitting | C22×#2380 | C3 |
| socle | C25 | C22×A4 |
| prime radical | C23×#127⋊C3 | C2 |
| cosocle | C2×C42 | C24×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C23 | |
5524 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 5524 conjugacy classes of subgroups — too many to draw. Only its 370 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,4)(9,11,20)(10,15,13)(12,16,24)(14,23,22)(17,18,19);(1,2)(3,4)(9,11)(10,19)(12,22)(13,21)(14,15)(16,20)(17,24)(18,23);(5,6,7,8)(11,19)(12,21)(13,20)(15,23)(16,24)(17,22);(1,2)136 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72214004, C2×#72213984, #72258949, … |
| maximal normal subgroup of | C2×#72214004, C2×#72213984, #72258949, … |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((3,4)(9,11,20)(10,15,13)(12,16,24)(14,23,22)(17,18,19), (1,2)(3,4)(9,11)(10,19)(12,22)(13,21)(14,15)(16,20)(17,24)(18,23), (5,6,7,8)(11,19)(12,21)(13,20)(15,23)(16,24)(17,22), (1,2));