Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#91904 |
| Sylow | C52 ⋊ GL(2,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C5 | D5×S4 |
| commutator, cosocle | C2×C5×A4 | C22×C5 |
| Frattini | C2 | C5×D5×S4 |
| Fitting, socle | C23×C52 | C2×D3 |
| hypocenter | C5×A4 | C5×D4 |
| LCS | G ▷ C2×C5×A4 ▷ C5×A4 | |
| UCS | C1 ◁ C2×C5 | |
314 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 314 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) = 60 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,4,2,3)(6,8)(9,10,11,12,13)(14,15,17,18,16);(2,3,4)(5,6)(7,8)(15,16)(17,18)170 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D5×GL(2,ℤ/4) |
| maximal normal subgroup of | C5×D5×GL(2,ℤ/4) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,3)(6,8)(9,10,11,12,13)(14,15,17,18,16), (2,3,4)(5,6)(7,8)(15,16)(17,18));