Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C22×C4 |
| direct product | C61×#29935 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×#121×C61 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×#117×C61 ◁ G | |
710 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,26,29)(21,24,27,30)(25,32,28,31)(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);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,22)(19,25)(21,27)(23,31)(26,28)(29,32);(3,7)(6,8)(10,14)(13,16)(17,19)(18,23)(20,26)(21,28)(22,29)(24,31)(25,27)(30,32);(1,3,4,7)(2,6,5,8)(18,22)(19,26)(24,30)(25,28)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 22923);