Finite Group DatabasebetaFinite Group Databasebeta| order | 15616 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C61 · Schur multiplier (H2) = C25 |
| direct product | C2×C61×#2095 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C61 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C61 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C61 ◁ C22×C42×C61 ◁ G | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 15616 |
A minimal faithful permutation representation — 4 generators on 83 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(7,8,10,12)(9,13,14,11)(15,16,18,21,20,17,19,22);(3,4)(5,6)(7,9,10,14)(8,11,12,13)(15,16)(17,20)(18,22)(19,21);(4,6)(15,17,20,16)(18,21,19,22)3904 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15616, 16715);