Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C59 · Schur multiplier (H2) = C22×C4 |
| direct product | C59×#8097 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C59 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C59 |
| Frattini, cosocle | C2×C42 | C23×C59 |
| socle | C22×C59 | C2×#95 |
| prime radical | C24×C59 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ C22×C42×C59 ◁ G | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 3 generators on 75 points:
(2,6)(3,4)(7,8)(9,10,13,15,14,11,12,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(1,2,4,8,5,7,3,6)(9,11,14,10)(12,15,13,16);(2,7)(6,8)3776 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 1085);