Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#8678 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #380 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C53×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,19)(12,24,26,16)(13,27)(14,18)(15,22)(17,20)(21,35,37,31)(23,32)(25,30,34,39)(28,36,38,40)(29,33)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,5)(6,7)(9,11,20,33,14,23,13,22)(10,15,27,32,18,29,17,19)(12,21,34,40,26,37,25,36)(16,28,39,35,24,38,30,31);(2,6)(4,8)(5,7)(9,12)(10,16)(11,21)(13,25)(14,26)(15,28)(17,30)(18,24)(19,31)(20,34)(22,36)(23,37)(27,39)(29,38)(32,35)(33,40)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1666);