Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C4 |
| direct product | C61×#7669 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C2×C42 | C23×C61 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C43×C61 ◁ G | |
384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 3 generators on 93 points:
(2,5,6,7)(9,10,13,16,14,11,12,15)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30)(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);(1,2,4,6)(3,7,8,5)(9,11,14,10)(12,15,13,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(9,12,14,13)(10,15,11,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 657);