Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13539 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,5)(4,6)(7,8)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,23,26,30)(21,24,28,29)(25,31,32,27)(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);(1,3,4,7)(2,5,6,8)(10,13)(11,16)(17,19,20,26)(18,23,22,30)(21,25,28,32)(24,31,29,27);(2,6)(5,8)(9,11,12,16)(10,14,13,15)(18,22)(19,25)(21,29)(23,27)(24,28)(26,32)(30,31)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6527);