Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#13162 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C2×D4 |
| commutator | C2×C4 | C22×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C8×D4 |
| prime radical | C23×C61 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ C22×C16×C61 ◁ G | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,4)(5,6,10,16,11,15,26,32,12,17,25,14,9,27,8,23)(7,13,20,28,21,29,18,33,22,30,34,31,19,35,24,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,4)(5,7,10,20,11,21,26,18,12,22,25,34,9,19,8,24)(6,13,16,28,15,29,32,33,17,30,14,31,27,35,23,36);(5,8,9,25,12,26,11,10)(6,14,15,23,17,16,27,32)(7,18,19,20,22,24,21,34)(13,28,29,33,30,31,35,36)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6150);