Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#12002 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C61 | #106 |
| commutator | C8 | C22×C8×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C2×C61 | C2×#334 |
| prime radical | C23×C61 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C61 ◁ C22×C8×C61 ◁ C2×C4×C8×C61 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,5,11,7,12,22,28,8,13,21,10,6,24,4,20)(3,14,17,26,19,29,16,25,15,27,23,32,18,31,30,9)(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);(1,3,4,16,6,18,21,17,8,15,22,30,7,19,5,23)(2,9,10,14,12,26,20,29,13,25,11,27,24,32,28,31);(3,15)(9,25)(14,27)(16,30)(17,23)(18,19)(26,32)(29,31)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4990);