Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C22×C4 |
| direct product | C59×#13248 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C59 | #1972 |
| commutator | C2×C8 | C22×C4×C59 |
| Frattini, cosocle | C22×C8 | C23×C59 |
| prime radical | C2×#135×C59 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C59 ◁ C22×C4×C59 ◁ C59×#405 ◁ G | |
394 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)(3,15)(4,22,26,52)(5,28)(6,11)(7,32)(8,14)(9,18)(10,25,39,46)(12,30)(13,29)(16,41)(17,33,49,62)(19,35)(20,42)(21,45)(23,53,55,37)(24,56,58,40)(27,60,57,38)(31,44)(34,50)(36,51)(43,48,63,54)(47,61,59,64)(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);(1,3,16,19,7,20,5,18,8,21,6,31,29,50,30,51)(2,9,12,42,13,35,11,15,14,36,28,34,32,44,41,45)(4,23,25,59,27,49,58,48,26,55,46,47,57,17,24,54)(10,37,52,63,40,62,60,64,39,53,22,43,56,33,38,61);(1,4,6,25,7,27,30,58,8,26,16,46,29,57,5,24)(2,10,12,22,13,40,11,38,14,39,28,52,32,56,41,60)(3,17,19,48,20,23,18,47,21,49,31,54,50,55,51,59)(9,33,34,61,35,37,45,63,36,62,42,64,44,53,15,43)2006 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 6236);