Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C8×C61 · Schur multiplier (H2) = C2×C42 |
| direct product | C61×#12539 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C16×C61 | C22 |
| commutator | C2 | C42×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle, prime radical | C23×C61 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C16×C61 ◁ G | |
478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 101 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 3 generators on 101 points:
(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(1,2,3,4)(9,11)(10,15)(12,20)(13,21)(14,19)(16,26)(17,27)(18,25)(22,33)(23,31)(24,32)(28,37)(29,35)(30,36)(34,39)(38,40);(5,6,7,8)(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)9760 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 5527);