Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#60876 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C61×#525 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×#117×C61 ◁ G | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 5 generators on 109 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,23,27)(19,24,30,39)(20,32,34,40)(21,26,22,35)(25,36,41,33)(28,42,44,47)(29,38,31,45)(37,46,48,43)(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);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,28)(21,29)(22,31)(23,30)(25,37)(26,38)(27,39)(32,42)(33,43)(34,44)(35,45)(36,46)(40,47)(41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,43)(30,44)(31,46)(32,35)(38,47)(39,48)(42,45);(2,8)(6,9)(7,16)(14,15)(17,21,23,22)(18,26,27,35)(19,29,30,31)(20,33,34,36)(24,38,39,45)(25,40,41,32)(28,43,44,46)(37,47,48,42);(19,30)(24,39)(28,44)(29,31)(37,48)(38,45)(42,47)(43,46)2989 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 53864);