Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42×C61 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C61×#2875 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C24 |
| commutator | C2 | C23×C42×C61 |
| Frattini, cosocle | C2×C4 | C25×C61 |
| socle | C23×C61 | C23×C4 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ G | |
3278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 83 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(7,8,10,14,11,12,17,19)(9,13,16,20,15,18,21,22);(3,4,5,6)(8,12)(13,18)(14,19)(20,22);(7,9)(8,13)(10,16)(11,15)(12,18)(14,20)(17,21)(19,22);(9,15)(13,18)(16,21)(20,22)8296 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53044);