Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#29778 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×C61×#466 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×#117×C61 ◁ G | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 89 points:
(6,9)(10,12)(13,14,18,22,17,21,16,20)(15,23,24,27,25,19,26,28)(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);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,17,25)(14,19,21,23)(16,26,18,24)(20,27,22,28);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(16,18)(19,27)(21,22)(23,28)(24,26);(7,11)(10,12)(13,16,17,18)(14,20,21,22)(15,24,25,26)(19,28,23,27)2440 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 22766);