Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#9332 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #386 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| prime radical | C53×#3194 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18)(19,26)(20,27,29,31)(21,24)(22,25)(23,28,32,30)(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);(1,3,5,10)(2,6)(4,9,12,15)(7,13)(8,14)(11,16)(17,19,26,18,22,24,21,25)(20,27,28,23,29,31,30,32);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,23)(19,27)(21,30)(22,29)(24,31)(25,32)(26,28)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 2320);