Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C61 · Schur multiplier (H2) = C2×C4 |
| direct product | C61×#12670 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C61 | D8 |
| commutator | C8 | C2×C42×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C22×C61 | C4×D8 |
| prime radical | C23×C61 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C61 ◁ C22×C8×C61 ◁ C2×C4×C8×C61 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 3904 |
A minimal faithful permutation representation — 3 generators on 97 points:
(5,6,10,16,11,17,9,15,12,18,26,34,25,33,8,14)(7,19,22,13,23,30,27,31,24,35,20,32,28,29,21,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,2,3,4)(5,7)(6,13)(8,20)(9,22)(10,27)(11,28)(12,24)(14,29)(15,30)(16,35)(17,36)(18,32)(19,34)(21,26)(23,25)(31,33);(5,8,25,26,12,9,11,10)(6,14,33,34,18,15,17,16)(7,20,23,21,24,22,28,27)(13,29,31,19,32,30,36,35)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5658);