Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#18519 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C2×#120 |
| prime radical | C2×#41×C61 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ G | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,23,28)(19,25,30,31)(20,24,29,32)(21,26,22,27)(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);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19,23,30)(18,24,28,32)(20,21,29,22)(25,27,31,26);(17,20)(18,25)(19,21)(22,30)(23,29)(24,27)(26,32)(28,31);(17,21,23,22)(18,26,28,27)(19,29,30,20)(24,25,32,31)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 11507);