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×#22808 |
| 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 | |
782 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)(13,14,17,20,18,21,25,16)(15,22,23,27,24,28,26,19)(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)(13,15,18,24)(14,19,21,27)(16,22,20,28)(17,26,25,23);(5,8,9,7)(6,10,11,12);(13,16,25,21,18,20,17,14)(15,19,26,28,24,27,23,22)4270 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 15796);