Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13005 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1874 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C53×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C2×#131×C53 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10,5,11,22,24,7,12,4,19,21,30,20,8)(3,13,16,31,17,9,25,26,18,27,14,23,15,28,32,29)(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);(2,8,12,24)(4,20)(5,21)(6,22)(9,23,28,31)(10,11,19,30)(13,29,27,26)(14,32)(15,17)(16,25);(1,3)(2,9)(4,16)(5,17)(6,14)(7,18)(8,23)(10,29)(11,27)(12,28)(13,30)(15,21)(19,26)(20,25)(22,32)(24,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5993);