Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C59 · Schur multiplier (H2) = C23×C4 |
| direct product | C59×#8200 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C59 | C2×D4 |
| commutator | C4 | C2×C4×C8×C59 |
| Frattini, cosocle | C2×C42 | C23×C59 |
| socle | C23×C59 | C4×D4 |
| prime radical | C22×D4×C59 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C59 ◁ C2×C4×C8×C59 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 79 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 3 generators on 79 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,16,19,17,15,18,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79);(2,4)(5,6,8,12,9,11,7,10);(6,11)(10,12)(13,15,16,20,17,14,18,19)5192 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 1188);