Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C25 |
| direct product | C61×#11089 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #393 |
| commutator | C24 | C22×C4×C61 |
| Frattini, cosocle | C25 | C23×C61 |
| prime radical | C23×D4×C61 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C25×C61 ◁ G | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 81 points:
(2,4)(5,6)(7,10)(8,15,17,11)(9,12)(13,20,16,18)(14,19)(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);(1,2,3,4)(5,7)(6,10)(8,16,17,13)(9,14)(11,18,15,20)(12,19);(5,8)(6,11)(7,13)(9,17)(10,18)(12,15)(14,16)(19,20)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4077);