Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C26 |
| direct product | C61×#10777 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | #379 |
| commutator | C2×C4 | C2×C42×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| prime radical | C24×C61 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C23×C4×C61 ◁ G | |
1178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(5,6,10,16,11,14,9,15)(7,17,19,30,20,12,25,27)(8,21,23,13,24,29,22,28)(18,32,33,26,34,36,35,31)(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);(1,2,3,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,17)(15,27)(16,30)(21,36)(22,33)(23,35)(24,34)(28,31)(29,32);(5,8,11,24)(6,13,14,28)(7,18,20,34)(9,23,10,22)(12,26,17,31)(15,29,16,21)(19,35,25,33)(27,32,30,36)3538 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 3765);