Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13030 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13030 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #1879 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C53×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ C53×#405 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,10,6,11,22,25,7,12,4,18,21,31,19,8)(3,13,15,30,16,32,23,9,17,26,14,27,24,29,20,28)(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);(2,8,12,25)(4,19)(5,22)(6,21)(9,13,28,26)(10,11,18,31)(14,20)(15,23)(16,24)(27,29,30,32);(1,3)(2,9)(4,20)(5,23)(6,24)(7,17)(8,13)(10,29)(11,30)(12,28)(14,19)(15,22)(16,21)(18,32)(25,26)(27,31)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6018);