Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#19600 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C2×#127 |
| prime radical | C61×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ G | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(18,22)(23,30)(24,29)(25,31);(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 12588);