Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C59 · Schur multiplier (H2) = C1 |
| direct product | C59×#7464 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C59 | #367 |
| commutator | C16 | C2×C8×C59 |
| Frattini, cosocle | C4×C16 | C22×C59 |
| socle | C2×C59 | #1372 |
| prime radical | C22×C59 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C59 ◁ C2×C8×C59 ◁ C4×C8×C59 ◁ C4×C16×C59 ◁ G | |
76 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 2 generators on 123 points:
(1,2,5,11,7,12,28,38,8,14,23,42,30,13,6,29)(3,15,18,44,20,46,19,56,21,48,27,45,57,47,58,64)(4,22,16,39,25,59,49,41,26,60,50,40,55,32,24,10)(9,31,33,62,35,63,61,17,37,51,34,53,36,52,43,54)(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);(1,3,16,31,6,19,4,17,30,57,24,52,23,18,55,62,8,21,50,51,28,58,26,54,7,20,49,63,5,27,25,53)(2,9,32,56,11,34,10,15,12,35,22,45,38,43,39,46,14,37,59,64,42,33,41,48,13,36,60,44,29,61,40,47)3068 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 452);