Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C8 |
| direct product | C53×#8270 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C53 | C2×D4 |
| commutator | C2×C4 | C2×C42×C53 |
| Frattini, cosocle | C2×C42 | C23×C53 |
| socle | C22×C53 | C4×C4⋊C4 |
| prime radical | C23×C53 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
342 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(18,22,24,27)(23,29,30,31)(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);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,18)(19,23)(20,27)(21,24)(22,28)(25,31)(26,30)(29,32);(2,7)(6,12)(8,14)(13,16)(17,19,20,25,21,26,28,32)(18,23,27,31,24,30,22,29)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 1258);