Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#11241 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C61 | #121 |
| commutator | C22×C4 | C22×C4×C61 |
| Frattini, cosocle | C23×C4 | C23×C61 |
| socle | C22×C61 | #380 |
| prime radical | C24×C61 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C61 ◁ C23×C4×C61 ◁ G | |
638 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)(7,15)(8,12,20,26)(9,21)(10,14)(11,17)(13,22)(16,27,30,36)(18,28)(19,32,33,25)(23,34,35,29)(24,31)(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)(5,7,9,17,10,18,22,31)(6,11,13,15,14,24,21,28)(8,16,19,29,20,30,33,34)(12,23,25,27,26,35,32,36);(2,4)(5,8,9,19,10,20,22,33)(6,12,13,25,14,26,21,32)(7,16,17,29,18,30,31,34)(11,23,15,27,24,35,28,36)3172 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 4229);