Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C61 · Schur multiplier (H2) = C28 |
| direct product | C61×#61771 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C61 |
| prime radical | C61×#3541 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C61×#525 ◁ G | |
1688 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)(19,24,29,38)(20,31,33,41)(21,26,22,35)(25,36,42,32)(28,43,44,48)(30,39,34,40)(37,45,47,46)(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)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24)(20,28)(21,34)(22,30)(23,29)(25,37)(26,40)(27,38)(31,43)(32,46)(33,44)(35,39)(36,45)(41,48)(42,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,41)(27,42)(29,44)(30,45)(31,35)(34,46)(38,47)(39,43)(40,48);(3,11)(6,9)(10,16)(13,15)(19,29)(24,38)(28,44)(30,34)(37,47)(39,40)(43,48)(45,46);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,26,27,35)(19,30,29,34)(20,32,33,36)(24,39,38,40)(25,41,42,31)(28,45,44,46)(37,43,47,48)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 54759);