Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C2×#41×C61 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C4×#41×C61 ◁ G | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,28,23,24)(20,31,34,39)(21,35,37,26)(25,32,41,29)(30,38,36,42)(33,46,47,44)(40,43,48,45)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,36,37,30)(25,39,41,31)(26,42,35,38)(33,43,47,45)(40,44,48,46);(3,10)(6,9)(11,16)(13,15)(17,20,22,34)(18,25,27,41)(19,29,23,32)(21,33,37,47)(24,31,28,39)(26,40,35,48)(30,43,36,45)(38,44,42,46);(2,7)(6,9)(8,14)(13,15)(17,21)(18,26)(19,30)(20,33)(22,37)(23,36)(24,38)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(39,46)(41,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(21,37)(26,35)(30,36)(33,47)(38,42)(40,48)(43,45)(44,46)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54001);