Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C61 · Schur multiplier (H2) = C25×C4 |
| direct product | C61×#20401 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | C22×C22⋊C4 |
| commutator | C22 | C22×C42×C61 |
| Frattini, cosocle | C22×C4 | C24×C61 |
| socle | C2×C61 | C2×C4×C22⋊C4 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C22×M16×C61 ◁ G | |
1444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2,5,9,6,10,17,23)(3,11,14,25,13,7,18,20)(4,8,16,22,15,21,28,31)(12,24,27,32,26,19,29,30)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,7,10,11)(4,12)(5,14)(6,13)(8,19,21,24)(9,20,23,25)(15,26)(16,27)(17,18)(22,30,31,32)(28,29);(1,4,5,16,6,15,17,28)(2,8,9,22,10,21,23,31)(3,12,14,27,13,26,18,29)(7,19,20,30,11,24,25,32);(4,15)(8,21)(12,26)(16,28)(19,24)(22,31)(27,29)(30,32)5002 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 13389);