Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | Dic5×C5⋊S4 |
| derived | C52×A4 ⋊ C2×C4 |
| Sylow | C52 ⋊ C4×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C5⋊S4 |
| commutator, hypocenter | C52×A4 | C2×C4 |
| Fitting, socle | C23×C52 | C2×D3 |
| prime radical | C2×C5×C5⋊S4 | C2 |
| cosocle | C2×C52×A4 | C22 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 ◁ C2 | |
262 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 38 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:
(1,2,3,4)(5,7,8,6)(10,11)(12,13)(14,15)(16,17);(7,8)(9,10,12,13,11)(15,16)(17,18)104 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊D5×C5⋊S4, #72260214, #72260224 |
| maximal normal subgroup of | C22⋊D5×C5⋊S4, #72260214, #72260224 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,2,3,4)(5,7,8,6)(10,11)(12,13)(14,15)(16,17), (7,8)(9,10,12,13,11)(15,16)(17,18));