Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8803 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#8804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #383 |
| commutator | C42 | C22×C4×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C23×C53 | #126 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
216 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)(3,10,13,6)(4,7)(5,8)(9,11)(12,15,16,14)(17,18)(19,26,28,30)(20,22)(21,25)(23,27,31,32)(24,29)(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);(1,3,11,16,5,13,4,12)(2,6,9,15,8,10,7,14)(18,22,25,24)(23,26,31,30);(2,7,8,9)(6,14,10,15)(17,19,20,27,21,28,29,32)(18,23,24,30,25,31,22,26)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1792);