Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#17769 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| prime radical | C53×#3511 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C4×#41×C53 ◁ G | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,10,6,8,4,9)(3,11,12,7,14,16,13,15)(17,18)(19,29)(20,27)(21,26)(22,25)(23,28)(24,30)(31,32)(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)(2,7)(4,12)(5,13)(6,14)(8,15)(9,16)(10,11)(17,19,23,21)(18,24,28,32)(20,30,22,31)(25,29,27,26);(2,8)(7,15)(9,10)(11,16)(17,20,23,22)(18,25,28,27)(19,30,21,31)(24,29,32,26);(1,4,6,5)(2,9,8,10)(3,12,14,13)(7,16,15,11)(17,21,23,19)(18,26,28,29)(20,30,22,31)(24,25,32,27)2597 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10757);