Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C59 · Schur multiplier (H2) = C22×C4 |
| direct product | C59×#13244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C59 | #466 |
| commutator | C2×C8 | C22×C4×C59 |
| Frattini, cosocle | C22×C8 | C23×C59 |
| prime radical | C23×C59 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C22×C4×C59 ◁ C59×#405 ◁ G | |
402 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)(5,6,11,18,13,19,12,37)(7,20,24,40,25,48,36,51)(8,26,29,58,31,59,30,45)(9,17,10,35,34,49,33,16)(14,38,41,22,43,23,42,32)(15,44,46,61,47,27,57,28)(21,39,54,64,55,66,62,65)(50,60,56,63,67,52,68,53)(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)(5,7,9,32,12,24,33,38,13,25,34,22,11,36,10,23)(6,14,35,20,18,42,49,51,19,43,16,48,37,41,17,40)(8,21,27,60,30,54,61,63,31,55,44,52,29,62,28,53)(15,39,58,50,46,65,59,68,47,66,45,67,57,64,26,56);(5,8,9,27,12,30,33,61,13,31,34,44,11,29,10,28)(6,15,35,58,18,46,49,59,19,47,16,45,37,57,17,26)(7,21,32,60,24,54,38,63,25,55,22,52,36,62,23,53)(14,39,20,50,42,65,51,68,43,66,48,67,41,64,40,56)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 6232);