Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C59 · Schur multiplier (H2) = C24 |
| direct product | C2×C59×#1377 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C59 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C59 |
| Frattini, cosocle | C22×C8 | C23×C59 |
| socle | C23×C59 | C22⋊C8 |
| prime radical | C24×C59 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C59 ◁ C23×C8×C59 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 7552 |
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);(7,8,9,13,10,14,16,20,11,12,15,18,17,19,21,22);(3,4,5,6)(8,12)(9,15)(14,19)(16,21)5192 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 5538);