Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#11492 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C2×C22⋊C8 |
| prime radical | C53×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,3,6,4,5,7,8)(9,10,14,20,16,19,12,18,15,21,29,37,13,30,28,38)(11,22,25,32,24,17,27,33,26,35,40,36,31,34,23,39)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(2,5)(3,7)(9,11)(10,17)(12,27)(13,31)(14,25)(15,26)(16,24)(18,36)(19,35)(20,33)(21,34)(22,30)(23,28)(29,40)(32,38)(37,39);(9,12,13,14,15,28,16,29)(10,18,30,20,21,38,19,37)(11,23,31,40,26,27,24,25)(17,32,22,39,34,36,35,33)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4480);