Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C59 · Schur multiplier (H2) = C25 |
| direct product | C59×#33458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | C22×D8 |
| commutator | C8 | C23×C4×C59 |
| Frattini, cosocle | C2×C8 | C24×C59 |
| prime radical | C23×C59 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ #117×C59 ◁ C59×#405 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2,4,3)(5,6)(7,17)(8,13)(9,25)(10,30)(11,14)(12,19)(15,28)(16,27)(18,42)(20,49)(21,43)(22,37)(23,52)(24,55)(26,40)(29,41)(31,39)(32,45)(33,48)(34,57)(35,54)(36,51)(38,47)(44,64)(46,67)(50,63)(53,62)(56,60)(58,68)(59,65)(61,66)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(5,7,11,21)(6,12,14,33)(8,18,22,44)(9,26,29,57)(10,31,27,20)(13,32,37,59)(15,39,38,49)(16,40,30,34)(17,41,43,25)(19,47,48,28)(23,53,50,68)(24,56,51,46)(35,60,61,67)(36,62,55,58)(42,63,64,52)(45,66,65,54);(1,3,4,2)(5,8,11,22)(6,13,14,37)(7,18,21,44)(9,23,29,50)(10,24,27,51)(12,32,33,59)(15,35,38,61)(16,36,30,55)(17,42,43,64)(19,45,48,65)(20,46,31,56)(25,52,41,63)(26,53,57,68)(28,54,47,66)(34,58,40,62)(39,60,49,67);(6,14)(8,22)(12,33)(15,38)(16,30)(17,43)(18,44)(23,50)(24,51)(25,41)(34,40)(39,49)(45,65)(46,56)(53,68)(54,66)3776 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 26446);