Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C53×#33468 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #442 |
| commutator | C8 | C23×C4×C53 |
| Frattini, cosocle | C2×C8 | C24×C53 |
| socle | C2×C53 | C2×#442 |
| prime radical | C53×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2)(3,14,18,30)(4,12)(5,11)(6,21)(7,23)(8,13)(9,17,26,15)(10,19)(16,28,24,31)(20,29)(22,25,32,27)(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)(2,9)(4,15)(5,16)(6,22)(7,24)(8,18)(10,14)(11,25)(12,28)(13,26)(17,20)(19,32)(21,30)(23,27)(29,31);(1,4,7,19,8,20,5,6)(2,10,11,12,13,21,23,29)(3,15,24,32,18,17,16,22)(9,14,25,28,26,30,27,31);(1,5,8,7)(2,11,13,23)(3,16,18,24)(4,6,20,19)(9,25,26,27)(10,12,21,29)(14,28,30,31)(15,22,17,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26456);