Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12833 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#12833 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1807 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C2×#41×C53 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,13)(3,14)(4,20)(5,15,25,31)(6,12,8,27)(9,10)(11,24,29,26)(16,22)(17,21)(18,32)(19,30)(23,28)(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);(1,3,5,16,8,18,26,20,7,19,25,10,6,17,24,28)(2,9,11,14,12,4,15,21,13,22,29,30,27,23,31,32);(1,4)(2,10)(3,15)(5,21)(6,9)(7,23)(8,22)(11,17)(12,28)(13,16)(14,24)(18,29)(19,31)(20,27)(25,32)(26,30)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5821);