Finite Group DatabasebetaFinite Group Databasebeta| order | 3600 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#14268434 |
| Fitting, socle | C2×C32×C52 ⋊ C2×C4 |
| derived, Hall | C32×C52 ⋊ C22×C4 |
| Sylow | C52 ⋊ C2×#624 |
| Sylow | C32 ⋊ C22×#1043 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #14268434 |
| commutator, hypocenter | C32×C52 | C22×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C32×C52 | C2×C4 |
| prime radical | C2×C3⋊D3×C5⋊D5 | C2 |
| cosocle | C32×C5⋊D5 | C23 |
| LCS | G ▷ C32×C52 | |
| UCS | C1 ◁ C2 | |
992 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 992 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(6,7,10,8,9)(11,12,13)(14,15,16)(17,18);(1,2,5,3,4)(12,13)(15,16);(2,3,4,5)(7,8,9,10)(15,16)108 conjugacy classes — view the full character table.
| maximal subgroup of | #624×#1043, #72231400, #72231569, … |
| maximal normal subgroup of | #624×#1043, #72231400, #72231569, … |
| maximal quotient of | #72231569, C22×#14268434, C2×#72229567 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((6,7,10,8,9)(11,12,13)(14,15,16)(17,18), (1,2,5,3,4)(12,13)(15,16), (2,3,4,5)(7,8,9,10)(15,16));