Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#33701 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C22×C53 | C22×D8 |
| prime radical | C53×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 93 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,24)(12,18)(13,20)(14,32)(15,34)(16,35)(17,23)(19,28)(21,33)(22,31)(25,39)(26,36)(27,37)(29,40)(30,38)(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);(1,3,4,7)(2,5,6,8)(9,11,15,26,16,27,14,12,17,28,33,29,13,25,31,30)(10,18,22,37,20,36,21,24,23,38,32,39,35,40,34,19);(2,6)(5,8)(9,12,14,25,13,26,15,28,17,30,31,27,16,29,33,11)(10,19,21,36,35,39,22,18,23,24,34,40,20,37,32,38);(9,13,17,16)(10,20,23,35)(11,25,28,27)(12,26,30,29)(14,15,31,33)(18,36,38,40)(19,37,24,39)(21,32,34,22)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26689);