Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C4 |
| direct product | C61×#12014 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | C2×D4 |
| commutator | C2×C4 | C22×C8×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C22×C61 | C2×C4⋊C8 |
| prime radical | C23×C61 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ C2×C4×C8×C61 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,20,24,21,25,30,36,22,26,31,37,32,38,42,46)(19,23,28,34,29,35,41,45,27,33,39,43,40,44,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,33)(30,41)(31,39)(32,40)(36,45)(37,43)(38,44)(42,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,27)(23,33)(28,39)(29,40)(34,43)(35,44)(41,47)(45,48)4636 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5002);