Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#9078 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C61 | C2×D4 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C2×C42 | C23×C61 |
| socle | C23×C61 | C2×C22⋊C4 |
| prime radical | C2×#121×C61 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C61 ◁ C22×C42×C61 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2)(3,4)(6,9)(7,10)(11,12)(13,14,15,17,16,19,20,18)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(5,6,10,12,8,11,7,9)(14,17,19,18);(2,4)(6,11)(9,12)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 2066);