Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#9770 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #390 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C23×C53 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,9)(4,13)(5,7)(6,11)(8,10)(12,15)(14,16)(17,18,21,25,22,27,32,26)(19,20,28,30,24,31,29,23)(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);(1,3,10,16,5,12,4,11)(2,6,13,15,7,14,8,9)(17,19,22,24)(18,23,27,30)(20,25,31,26)(21,29,32,28);(2,7)(6,14)(8,13)(9,15)(17,20,21,30,22,31,32,23)(18,24,26,28,27,19,25,29)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2758);