Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C61 · Schur multiplier (H2) = C4 |
| direct product | C61×#7198 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C61 | C22⋊C4 |
| commutator | C2×C4 | C4×C8×C61 |
| Frattini, cosocle | C2×C4×C8 | C22×C61 |
| socle | C22×C61 | #336 |
| prime radical | C23×C61 | #97 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
192 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 2 generators on 109 points:
(1,2)(3,9,10,6)(4,8)(5,7)(11,15,16,13)(12,14)(17,18,21,26,22,27,36,44,23,28,37,45,20,25,34,42)(19,29,31,46,30,40,47,43,32,41,35,48,33,24,38,39)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,3,4,11,5,10,12,16)(2,6,8,13,7,9,14,15)(17,19)(18,24)(20,33)(21,35)(22,30)(23,32)(25,41)(26,43)(27,29)(28,40)(31,37)(34,47)(36,38)(39,45)(42,46)(44,48)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 186);