Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×A5 |
| commutator, hypocenter | C3×SL(2,𝔽5) | C23 |
| Fitting | C3×D4 | C2×A5 |
| socle, cosocle | C2×C3 | C23×A5 |
| hypercenter | D4 | D3×A5 |
| LCS | G ▷ C3×SL(2,𝔽5) | |
| UCS | C1 ◁ C2 ◁ D4 | |
742 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 742 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 3 generators on 51 points:
(4,17,9,35,47,41,45,46,23,34,7,8,19,40,49,28,36,48,10,18)(5,30,11,42)(6,20,13,51,37,38,43,26,15,16,14,29,27,50,22,44,31,12,21,32)(24,39,33,25);(1,2)(4,10,39,45,49,22,27,5,6,15,7,23,25,36,47,37,13,11,14,21)(8,34,32,26,30,44,50,41,40,33,17,18,16,12,42,38,51,28,35,24)(9,31,19,43)(20,46,29,48);(1,2,3)(4,30,7,42)(5,28,11,41)(6,51,14,50)(8,47,17,49)(9,20,19,29)(10,26,23,12)(13,40,27,35)(15,38,21,44)(16,39,32,25)(18,45,34,36)(22,24,37,33)(31,48,43,46)72 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((4,17,9,35,47,41,45,46,23,34,7,8,19,40,49,28,36,48,10,18)(5,30,11,42)(6,20,13,51,37,38,43,26,15,16,14,29,27,50,22,44,31,12,21,32)(24,39,33,25), (1,2)(4,10,39,45,49,22,27,5,6,15,7,23,25,36,47,37,13,11,14,21)(8,34,32,26,30,44,50,41,40,33,17,18,16,12,42,38,51,28,35,24)(9,31,19,43)(20,46,29,48), (1,2,3)(4,30,7,42)(5,28,11,41)(6,51,14,50)(8,47,17,49)(9,20,19,29)(10,26,23,12)(13,40,27,35)(15,38,21,44)(16,39,32,25)(18,45,34,36)(22,24,37,33)(31,48,43,46));