Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C59 · Schur multiplier (H2) = C24 |
| direct product | C2×C59×#1386 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C59 | #102 |
| commutator | C8 | C2×C42×C59 |
| Frattini, cosocle | C22×C8 | C23×C59 |
| prime radical | C24×C59 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C59 ◁ C23×C4×C59 ◁ C23×C8×C59 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 3 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);(3,4,5,6)(7,8,11,15)(9,10,17,19)(12,14,20,22)(13,21,18,16);(7,9,12,16,13,14,19,15,11,17,20,21,18,22,10,8)3776 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 5554);