Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C23×C4 |
| direct product | C61×#10810 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C61 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C2×C61 | #1434 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C61 ◁ C2×#117×C61 ◁ G | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,12,16,27)(4,17)(5,10)(6,20)(7,11)(8,14,23,15)(9,18)(13,22,29,32)(19,24)(21,25)(26,30,31,28)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,8,11,23)(4,13)(5,14)(6,15)(7,16)(9,22,25,32)(10,12,20,27)(17,31,24,26)(18,28)(19,29)(21,30);(1,4,5,18,7,19,6,21)(2,9,10,24,11,25,20,17)(3,13,14,28,16,29,15,30)(8,22,12,26,23,32,27,31)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3798);