Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#22769 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C61 | C22×D4 |
| commutator | C2×C4 | C23×C4×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C61 ◁ C4×C4⋊C4×C61 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 89 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(14,19)(16,25)(17,23)(20,28)(21,26)(22,27)(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);(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,26)(17,27)(18,22)(19,23)(21,24)(25,28);(5,8,9,7)(6,10,11,12)(13,15,18,24)(14,20,22,21)(16,23,25,17)(19,28,27,26);(13,16,18,25)(14,21,22,20)(15,23,24,17)(19,28,27,26)4270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 15757);