Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C22 |
| direct product | C61×#11973 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#102 |
| commutator | C2×C8 | C22×C4×C61 |
| Frattini, cosocle | C4×C8 | C23×C61 |
| prime radical | C23×C61 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ C2×C4×C8×C61 ◁ G | |
288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 109 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 109 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,22,27,20,29,36,26,23,28,39,44,37,25,21,38)(19,30,33,42,35,43,48,46,34,45,47,24,31,41,32,40)(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)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,34)(18,24,28,42)(20,31,37,35)(21,33,36,47)(22,32,39,48)(25,40,29,46)(26,43,38,41)(27,45,44,30);(2,7)(6,9)(8,15)(14,16)(17,20,23,37)(18,25,28,29)(19,31,34,35)(21,22,36,39)(24,40,42,46)(26,27,38,44)(30,43,45,41)(32,47,48,33)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4961);