Finite Group DatabasebetaFinite Group Databasebeta| order | 3625 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C52×C29 · Schur multiplier (H2) = C52 |
| direct product | C29×H(3,𝔽5) |
| unambiguous semidirect product | C52⋊(C5×C29) |
| unambiguous semidirect product | (C52×C29)⋊C5 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C5×C29 | C52 |
| commutator, Frattini, cosocle | C5 | C52×C29 |
| LCS | G ▷ C5 ▷ C1 | |
| UCS | C1 ◁ C5×C29 ◁ G | |
30 conjugacy classes of subgroups.
| minimal degree | μ(G) = 54 |
| transitive degree | μt(G) = 725 |
A minimal faithful permutation representation — 2 generators on 54 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29)(30,31,33,37,44)(32,35,39,46,51)(34,40,49,54,50)(36,42,48,45,52)(38,47,53,43,41);(31,34,41,42,35)(33,38,39,40,48)(37,45,49,46,47)(44,51,52,53,54)841 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3625, 3);