Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#17342 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C61 | C24 |
| commutator | C22 | C22×C42×C61 |
| Frattini, cosocle | C42 | C24×C61 |
| socle | C22×C61 | C22×C42 |
| prime radical | C2×#41×C61 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42×C61 ◁ G | |
688 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,6)(3,9)(8,13)(10,15)(17,19,20,26,21,25,27,31)(18,22,24,29,23,28,30,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(19,25)(22,28)(26,31)(29,32);(3,9)(7,12)(10,15)(14,16)(18,23)(22,28)(24,30)(29,32)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 10330);