Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22284 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C24×C4 |
| direct product | C53×#22286 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C53 | #2328 |
| commutator | C2×C4 | C23×C4×C53 |
| Frattini, cosocle | C22×C4 | C24×C53 |
| prime radical | C53×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C53 ◁ C22×C4×C53 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,8,12)(3,7,11,6)(4,15)(5,20,23,14)(9,17)(10,22,26,21)(13,19)(16,29)(18,25)(24,32)(27,31)(28,30)(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);(1,3,6,2,8,11,7,12)(4,9,18,13,19,25,17,15)(5,14,21,10,23,20,22,26)(16,24,31,27,32,29,30,28);(1,4)(2,9,12,25)(3,13,11,15)(5,16)(6,17)(7,18)(8,19)(10,24,26,29)(14,27,20,28)(21,30)(22,31)(23,32);(1,5,7,22,8,23,6,21)(2,10,3,14,12,26,11,20)(4,16,18,31,19,32,17,30)(9,24,13,27,25,29,15,28)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 15274);