Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#17993 |
| 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 | C53×#466 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C53×#431 ◁ G | |
576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,5,3,6,8,14,10)(4,11,13,16,12,7,15,9)(17,18,22,27)(19,25,21,31)(20,24,30,26)(23,28,32,29)(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);(4,12)(7,11)(9,16)(13,15)(17,19,22,21)(18,23,27,32)(20,28,30,29)(24,25,26,31);(1,3,5,8,6,10,14,2)(4,9,13,11,12,16,15,7)(17,20,22,30)(18,24,27,26)(19,28,21,29)(23,25,32,31);(1,4)(2,7)(3,9)(5,13)(6,12)(8,11)(10,16)(14,15)(17,21,22,19)(18,25,27,31)(20,28,30,29)(23,26,32,24)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 10981);