Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C59 · Schur multiplier (H2) = C22 |
| direct product | C59×#11617 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C59 | C22⋊C4 |
| commutator | C2×C4 | C22×C8×C59 |
| Frattini, cosocle | C4×C8 | C23×C59 |
| socle | C22×C59 | C2×C22⋊C8 |
| prime radical | C2×#41×C59 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C59 ◁ C2×C4×C8×C59 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,5,8,6,3,4,7)(9,10,12,17,14,18,21,24,13,15,20,22,11,16,19,23)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(1,3)(2,6)(4,8)(5,7)(10,15)(12,20)(16,18)(19,21);(1,4,6,5)(2,7,3,8)(9,11,13,14)(10,16,15,18)(12,19,20,21)(17,23,22,24)4484 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 4605);