Finite Group DatabasebetaFinite Group Databasebeta| order | 2400 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×S5 |
| commutator, cosocle, hypocenter | C5×SL(2,𝔽5) | C22 |
| Fitting, socle | C2×C5 | C2×S5 |
| LCS | G ▷ C5×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 | |
139 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has 139 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,4,2,5,3)(6,35,39,20,11,40,8,21,26,31,18,27)(7,30,24,9,10,32,14,33,13,16,17,15)(12,19,36,45,46,37,29,34,22,51,48,23)(25,52,49,41,47,44,38,42,28,43,53,50);(1,3)(4,5)(6,20,38,40,39,43)(7,16,11,33,17,35)(8,31,25,27,26,41)(9,18,30,10,21,14)(12,32,48,19,24,45)(13,51,29,15,46,34)(22,50,49,37,47,52)(23,53,42,36,44,28)48 conjugacy classes — view the full character table.
| maximal subgroup of | #72277237, #72277310 |
| maximal normal subgroup of | #72277310 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,4,2,5,3)(6,35,39,20,11,40,8,21,26,31,18,27)(7,30,24,9,10,32,14,33,13,16,17,15)(12,19,36,45,46,37,29,34,22,51,48,23)(25,52,49,41,47,44,38,42,28,43,53,50), (1,3)(4,5)(6,20,38,40,39,43)(7,16,11,33,17,35)(8,31,25,27,26,41)(9,18,30,10,21,14)(12,32,48,19,24,45)(13,51,29,15,46,34)(22,50,49,37,47,52)(23,53,42,36,44,28));