Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#29632 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×C61×#535 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×#117×C61 ◁ G | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,22)(17,21)(18,20)(19,25)(24,28)(26,27)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(6,9)(10,12)(15,24)(16,17)(19,27)(21,22)(23,28)(25,26);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,26)(14,19,20,28)(16,24,17,25)(21,27,22,23);(7,11)(10,12)(14,20)(19,28)(21,22)(23,27)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 22620);