Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C61 · Schur multiplier (H2) = C24 |
| direct product | C61×#13190 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C61 | C23 |
| commutator | C23 | C22×C8×C61 |
| Frattini, cosocle | C22×C8 | C23×C61 |
| socle | C23×C61 | C22×C8 |
| prime radical | C22×D4×C61 | C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C61 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 7808 |
A minimal faithful permutation representation — 3 generators on 85 points:
(2,4)(6,8)(9,10,14,18,15,17,22,23,13,11,16,20,12,19,21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,2,3,4)(9,11,14,20,15,19,22,24,13,10,16,18,12,17,21,23);(5,6,7,8)(9,12,13,15)(10,17,11,19)(14,21,16,22)(18,23,20,24)5368 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 6178);