Finite Group DatabasebetaFinite Group Databasebeta| order | 27830 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C23⋊((C11⋊C5)⋊D11) |
| unambiguous semidirect product | (C11⋊C5)⋊D253 |
| Fitting, socle, derived, Hall | C112×C23 ⋊ C2×C5 |
| Hall | C11×C23×C11⋊C5 ⋊ C2 |
| Hall | C11⋊D253 ⋊ C5 |
| Hall | C11×C11⋊C5 ⋊ D23 |
| Sylow | C112 ⋊ C5×D23 |
| Sylow | C23 ⋊ (C11⋊C5)⋊D11 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C112×C23 | C2×C5 |
| LCS | G ▷ C112×C23 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1265 |
A minimal faithful permutation representation — 2 generators on 45 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,29,30,28,33,26,27,34,31,32)(36,37)(38,39)(40,41)(42,43)(44,45);(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,25,28,33,27,30,34,29,32,31,26)(35,36,38,40,42,44,45,43,41,39,37)Polycyclic presentation:
2,5,11^2,23|c^a=c^10,d^a=d^10,e^a=e^22,c^b=c^4893 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(27830, 36);