Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#11620 |
| 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 | #121×C53 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 93 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 93 points:
(1,2,4,6,5,3,8,7)(9,10,13,19,15,17,29,33,14,18,28,34,12,24,26,37)(11,20,22,35,21,30,36,39,23,16,27,31,25,32,38,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,25)(13,27)(14,23)(15,21)(17,32)(18,20)(19,35)(22,28)(24,30)(26,36)(29,38)(31,34)(33,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12,14,15)(10,17,18,24)(11,21,23,25)(13,26,28,29)(16,30,20,32)(19,33,34,37)(22,36,27,38)(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, 4608);