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×#22055 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C59 | C22×D4 |
| commutator | C2×C4 | C23×C4×C59 |
| Frattini, cosocle | C22×C4 | C24×C59 |
| prime radical | C2×C59×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C59 ◁ C2×#117×C59 ◁ G | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 4 generators on 87 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14)(15,25)(16,20)(17,24)(18,23)(19,22)(21,27)(26,28)(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);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,18)(17,27)(19,26)(21,22)(23,25)(24,28);(7,12)(9,11)(13,16,18,15)(14,21,23,28)(17,27,19,26)(20,24,25,22);(6,10)(9,11)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 15043);