Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#11950 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #102 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| socle | C22×C61 | C2×#102 |
| prime radical | C23×C61 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18,21,26,23,28,36,25,22,27,37,44,35,43,20,34)(19,29,31,40,33,42,47,46,32,45,48,24,38,39,30,41)(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)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,32)(18,24,27,40)(20,31,36,48)(21,30,37,47)(23,38,35,33)(25,42,34,39)(26,45,44,29)(28,46,43,41);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4938);