Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13033 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1879 |
| 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 | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,12,15,19,6,11,9,3,14,18,8,13,5,17)(4,20,23,10,27,28,21,31,25,30,26,16,24,32,22,29)(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);(1,3,15,19,9,2,8,13)(4,16,24,29,25,10,27,31)(5,18,7,11,6,12,14,17)(20,21,28,23,30,22,32,26);(1,4)(2,10)(3,16)(5,26)(6,23)(7,21)(8,27)(9,25)(11,28)(12,30)(13,31)(14,22)(15,24)(17,32)(18,20)(19,29)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6024);