Finite Group DatabasebetaFinite Group Databasebeta| order | 14400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23×C3×C5 |
| direct product | C5⋊D5×A4⋊S4 |
| Fitting, socle | C24×C52 ⋊ C2×C3⋊D3 |
| derived | C52×A42 ⋊ C22 |
| Sylow | C52 ⋊ C2×A4⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C52×A42 | C22 |
| Fitting, socle | C24×C52 | C2×C3⋊D3 |
| LCS | G ▷ C52×A42 | |
| UCS | C1 | |
2696 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 2696 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 300 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,3,2)(5,8,6)(10,11)(12,13)(15,16)(17,18);(3,4)(7,8)(14,15,17,18,16);(6,7,8)(9,10,12,13,11)196 conjugacy classes — view the full character table.
| maximal subgroup of | #1043×A4⋊S4, #72260294, #72260445 |
| maximal normal subgroup of | #1043×A4⋊S4, #72260294, #72260445 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2)(5,8,6)(10,11)(12,13)(15,16)(17,18), (3,4)(7,8)(14,15,17,18,16), (6,7,8)(9,10,12,13,11));