Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#7162 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C61 | #102 |
| commutator | C8 | C4×C8×C61 |
| Frattini, cosocle | C2×C4×C8 | C22×C61 |
| socle | C2×C61 | #1277 |
| prime radical | C23×C61 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C61 ◁ C22×C8×C61 ◁ C2×C4×C8×C61 ◁ G | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 2 generators on 93 points:
(1,2,4,11,7,13,22,30,8,14,23,12,6,24,5,10)(3,15,17,28,20,26,32,31,21,29,16,9,19,25,18,27)(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);(1,3,5,18,6,19,23,16,8,21,22,32,7,20,4,17)(2,9,12,26,13,27,10,29,14,28,11,25,24,31,30,15)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 150);