Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C24 |
| direct product | C59×#13020 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | #447 |
| commutator | C2×C8 | C22×C4×C59 |
| Frattini, cosocle | C22×C8 | C23×C59 |
| prime radical | C23×C59 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C59×#405 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 3 generators on 127 points:
(2,4)(5,6,11,18,35,51,36,17,13,20,9,16,12,19,10,37)(7,21,25,44,39,42,24,14,27,43,23,53,26,50,38,45)(8,28,31,49,40,46,34,48,33,52,29,15,32,47,30,54)(22,56,59,55,63,68,58,64,61,67,57,65,60,41,62,66)(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);(1,2,3,4)(5,7)(6,14)(8,22)(9,24)(10,25)(11,38)(12,39)(13,27)(15,41)(16,50)(17,21)(18,42)(19,53)(20,45)(23,36)(26,35)(28,64)(29,58)(30,59)(31,62)(32,63)(33,61)(34,57)(37,43)(40,60)(44,51)(46,55)(47,65)(48,56)(49,68)(52,66)(54,67);(5,8,13,33)(6,15,20,49)(7,22,27,61)(9,34,11,30)(10,31,36,29)(12,40,35,32)(14,41,45,68)(16,28,18,52)(17,46,37,47)(19,54,51,48)(21,55,43,65)(23,58,25,62)(24,57,38,59)(26,63,39,60)(42,66,50,64)(44,56,53,67)2714 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 6008);