Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C53×#3155 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
454 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)(3,14)(4,8)(5,22)(6,11)(7,13)(9,17)(10,20)(12,21)(15,28)(16,31)(18,29)(19,32)(23,26)(24,30)(25,27)(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)(3,15)(4,20)(5,6)(9,14)(10,13)(12,22)(16,24)(17,18)(19,23)(21,25)(26,29)(28,31)(30,32);(1,3,4,16,7,19,20,18)(2,9,10,28,13,30,8,26)(5,23,6,24,21,15,25,17)(11,29,22,32,27,31,12,14)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6405);