Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#11633 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C4×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)(3,7)(4,6)(5,8)(9,10,13,19,15,20,29,36,14,18,28,34,12,17,26,33)(11,21,23,35,22,32,37,40,24,16,27,31,25,30,38,39)(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);(2,6)(3,4)(7,8)(9,11)(10,16)(12,25)(13,27)(14,24)(15,22)(17,32)(18,21)(19,35)(20,30)(23,28)(26,37)(29,38)(31,34)(33,39)(36,40);(1,3,5,4)(2,7,8,6)(9,12,14,15)(10,17,18,20)(11,22,24,25)(13,26,28,29)(16,30,21,32)(19,33,34,36)(23,37,27,38)(31,39,35,40)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4621);