Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13247 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1972 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×C53×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,12)(3,14,17,6)(4,19)(5,18,25,11)(8,16,9,13)(10,27)(15,26)(20,24)(21,30)(22,29)(23,31)(28,32)(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);(1,3,14,12,8,18,5,16,7,17,6,2,9,11,25,13)(4,20,22,28,23,31,32,29,24,19,27,30,26,15,21,10);(1,4,6,22,8,23,25,32,7,24,14,27,9,26,5,21)(2,10,11,19,13,28,3,15,12,29,18,20,16,30,17,31)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6235);