Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#8118 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #95 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C2×#95 |
| prime radical | C24×C53 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C42×C53 ◁ G | |
522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18)(19,27,29,23)(20,24)(21,26)(22,25)(28,32,30,31)(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)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,21,30,22,29,20,28)(18,23,26,32,25,27,24,31);(17,20,22,21)(18,24,25,26)(19,28,29,30)(23,31,27,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1106);