Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C59 · Schur multiplier (H2) = C22×C4 |
| direct product | C59×#12642 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C59 | C2×D8 |
| commutator | C8 | C2×C42×C59 |
| Frattini, cosocle | C22×C8 | C23×C59 |
| socle | C22×C59 | C4×D8 |
| prime radical | C2×#135×C59 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ C22×C8×C59 ◁ C2×C4×C8×C59 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 95 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 3 generators on 95 points:
(1,2)(3,4)(5,6,9,16,12,17,11,27)(7,18,21,31,24,36,23,29)(8,14,25,33,26,34,10,15)(13,28,30,19,32,22,35,20)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95);(2,4)(5,7,10,22,11,23,26,19,12,24,25,28,9,21,8,20)(6,13,14,31,16,30,33,36,17,32,34,29,27,35,15,18);(5,8,9,25,12,26,11,10)(6,14,16,33,17,34,27,15)(7,19,21,22,24,20,23,28)(13,29,30,18,32,31,35,36)3776 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 5630);