Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C59 · Schur multiplier (H2) = C29 |
| direct product | C2×C59×#3041 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C59 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C59 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C59 ◁ C2×#117×C59 ◁ G | |
3074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 5 generators on 81 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(4,6)(7,8)(9,12)(10,13)(11,15)(14,18)(16,20)(17,19)(21,22);(3,4,5,6)(7,9)(8,12)(10,17)(11,16)(13,19)(14,21)(15,20)(18,22);(8,13)(10,18)(12,19)(14,15)(17,22)(20,21);(9,16)(12,20)(17,22)(19,21)3776 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 53404);