Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#11349 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | #391 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C2×C61 | #1445 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C23×C4×C61 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,7,12,22)(4,14,16,29)(5,10)(6,9)(8,15,24,31)(11,26)(13,21,17,25)(18,19)(20,30)(23,28)(27,32)(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);(1,3,5,17,6,12,18,13)(2,7,19,21,9,22,10,25)(4,11,15,30,16,27,31,28)(8,23,29,32,24,20,14,26);(1,4,5,15,6,16,18,31)(2,8,19,29,9,24,10,14)(3,11,17,30,12,27,13,28)(7,20,21,26,22,23,25,32)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4337);