Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19328 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#19344 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3169 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31)(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);(17,18,20,25,22,26,21,23)(19,24,27,31,29,32,28,30);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,29)(18,24,26,32)(20,27,21,28)(23,30,25,31);(2,7)(6,15)(8,12)(9,16)(17,20,22,21)(18,25,26,23)(19,28,29,27)(24,30,32,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 12332);