Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C24×C4 |
| direct product | C61×#25922 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C61 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| prime radical | C2×#127×C61 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C61 ◁ C2×C42×C61 ◁ G | |
1072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1952 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,17,18,24)(13,21,19,16)(15,23,22,20)(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);(1,3,5,4)(2,6,8,7)(10,14)(11,18)(13,19)(15,22);(1,3,5,4)(2,7,8,6)(9,11,19,23,12,18,13,20)(10,15,21,24,14,22,16,17);(1,4,5,3)(2,7,8,6)(10,14)(15,22)(16,21)(17,24)2623 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 18910);