Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#17953 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C22×D4 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C42 | C24×C53 |
| socle | C22×C53 | C2×C4×D4 |
| prime radical | C23×C53 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C53×#431 ◁ G | |
558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,21,24,22,25,20,30)(19,23,29,31,27,26,28,32)(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);(3,9)(6,12)(10,15)(13,16)(17,18,22,25)(19,26,27,23)(20,30,21,24)(28,31,29,32);(1,3,4,10,5,9,11,15)(2,6,8,13,7,12,14,16)(17,19,20,28,22,27,21,29)(18,23,30,32,25,26,24,31);(2,7)(6,12)(8,14)(13,16)(19,27)(23,26)(28,29)(31,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10941);