Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C61 · Schur multiplier (H2) = C28 |
| direct product | C2×C61×#2980 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C24 |
| commutator | C22 | C24×C4×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C23×C61 | C23×C4 |
| prime radical | C2×C61×#532 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ G | |
2186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 87 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87);(3,4)(5,9)(6,8)(7,10)(11,12,14,18,15,19,22,26)(13,16,21,23,20,17,24,25);(3,5)(4,7)(6,10)(8,9)(11,13)(12,16)(14,21)(15,20)(17,19)(18,23)(22,24)(25,26);(5,10)(7,9)(11,13)(12,17)(14,21)(15,20)(16,19)(18,25)(22,24)(23,26);(4,8)(7,9)(13,20)(16,17)(21,24)(23,25)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53216);