Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13234 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1964 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| prime radical | C53×#3058 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#131×C53 ◁ G | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,11,7,12,19,10)(3,13,16,14,17,5,9,23)(4,18,21,8,22,25,32,26)(15,29,30,27,31,28,24,20)(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)(2,7,17,23)(4,15,28,26)(5,10,19,9)(6,16,13,11)(8,22,31,29)(18,32,24,20)(21,30,27,25);(1,4,19,32,7,22,6,21)(2,8,11,25,12,26,10,18)(3,15,9,24,17,31,16,30)(5,20,23,29,13,27,14,28)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6222);