Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×(C5⋊S4)⋊D5 |
| Fitting, socle | C24×C52 ⋊ C2×C3×D3 |
| derived | C22×C52×A4 ⋊ C22×C3 |
| Sylow | C52 ⋊ C2×A4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C52×A4 | C22×C3 |
| Fitting, socle | C24×C52 | C2×C3×D3 |
| LCS | G ▷ C22×C52×A4 | |
| UCS | C1 | |
1664 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 1664 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,4,3)(5,7,6)(10,11)(12,13)(14,15)(16,17);(1,4)(2,3)(7,8)(9,10,12,13,11)(15,16)(17,18)184 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#72214019, #72260294, #72260373 |
| maximal normal subgroup of | A4×#72214019, #72260294, #72260373 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,4,3)(5,7,6)(10,11)(12,13)(14,15)(16,17), (1,4)(2,3)(7,8)(9,10,12,13,11)(15,16)(17,18));