Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#25893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C61 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×C61×#550 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C2×C4×D4×C61 ◁ G | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 89 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,27)(17,24)(18,23)(19,22)(20,26)(25,28)(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);(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,19)(17,25)(18,26)(21,23)(22,28)(24,27);(1,2,3,4)(5,7,9,8)(6,11,12,10)(13,16,17,26,18,25,19,15)(14,21,22,28,23,20,24,27);(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,22,23,24)(15,25,26,16)(20,28,21,27)2806 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 18881);