Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#10961 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | C2×C22⋊C4 |
| commutator | C23 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | C4×C22⋊C4 |
| prime radical | C22×#41×C61 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C23×C8×C61 ◁ G | |
926 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2)(3,4)(5,6,9,14,10,13,22,26)(7,11,17,24,16,25,20,29)(8,12,19,27,18,28,21,30)(15,31,32,36,33,23,34,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(2,4)(5,7)(6,11,13,25)(8,15)(9,20)(10,16)(12,23,28,31)(14,29,26,24)(17,22)(18,33)(19,34)(21,32)(27,36,30,35);(5,8,9,21,10,18,22,19)(6,12,26,27,13,28,14,30)(7,15,20,32,16,33,17,34)(11,23,24,36,25,31,29,35)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3949);