Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C59 · Schur multiplier (H2) = C23×C4 |
| direct product | C59×#22565 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C59 | C22×D4 |
| commutator | C2×C4 | C23×C4×C59 |
| Frattini, cosocle | C22×C4 | C24×C59 |
| prime radical | C22×#41×C59 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C59 ◁ C4×C22⋊C4×C59 ◁ G | |
966 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(14,19)(15,17)(20,27)(21,25)(22,26)(23,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(5,7,8,11)(6,9,10,12)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(6,10)(9,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23)3422 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 15553);