Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C59 · Schur multiplier (H2) = C25 |
| direct product | Q8×#104×C59 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C59 | C22×D4 |
| commutator | C2×C4 | C23×C4×C59 |
| Frattini, cosocle | C22×C4 | C24×C59 |
| socle | C22×C59 | C22×C22⋊C4 |
| prime radical | C2×#41×C59 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C59 ◁ C2×C4×Q8×C59 ◁ G | |
1096 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 3776 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2,4,6)(3,7,8,5)(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);(1,3,4,8)(2,5,6,7);(9,10,12,15,13,11,16,14);(9,11)(10,13)(12,14)(15,16)4130 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 15516);