Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C23×C4 |
| direct product | C53×#33697 |
| 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 | |
598 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)(11,19)(12,18)(13,20)(14,22)(15,21)(16,32)(17,23)(24,34)(25,36)(26,40)(27,37)(28,33)(29,35)(30,38)(31,39)(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)(9,11,15,25,16,26,31,29,17,27,30,28,13,24,14,12)(10,18,22,34,20,33,38,37,23,35,39,40,32,36,21,19);(2,6)(5,8)(9,12,14,24,13,28,30,27,17,29,31,26,16,25,15,11)(10,19,21,36,32,40,39,35,23,37,38,33,20,34,22,18);(9,13,17,16)(10,20,23,32)(11,24,27,26)(12,28,29,25)(14,30,31,15)(18,33,35,36)(19,34,37,40)(21,22,38,39)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26685);