Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C22 |
| direct product | C59×#13636 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | C2×D16 |
| commutator | C16 | C22×C4×C59 |
| Frattini, cosocle | C2×C16 | C23×C59 |
| socle | C2×C59 | C2×#366 |
| prime radical | C23×C59 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C59 ◁ C22×C4×C59 ◁ C22×C8×C59 ◁ C22×C16×C59 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 3 generators on 123 points:
(1,2,5,11,8,13,7,29)(3,14,18,40,16,38,20,51)(4,21,23,52,25,54,24,55)(6,27,26,37,28,56,22,12)(9,30,32,45,31,48,57,17)(10,34,35,44,36,43,53,15)(19,50,49,61,46,64,47,41)(33,42,39,58,59,62,60,63)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3,15,42,6,19,23,30,5,18,34,58,26,49,25,45,8,16,44,62,28,46,24,48,7,20,43,63,22,47,4,17)(2,9,21,41,12,33,53,51,29,57,55,64,56,60,36,38,13,31,54,61,37,59,35,40,11,32,52,50,27,39,10,14);(3,16)(9,31)(14,38)(17,45)(18,20)(19,46)(30,48)(32,57)(33,59)(39,60)(40,51)(41,61)(42,62)(47,49)(50,64)(58,63)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 6624);