Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C26 |
| direct product | C53×#10757 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #386 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C23×C4 | C23×C53 |
| socle | C2×C53 | C22⋊#95 |
| prime radical | C22×#41×C53 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C23×C4×C53 ◁ G | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,10,6,9,18,11)(3,7,14,21,13,20,15,25)(4,8,17,22,16,24,30,23)(12,26,28,32,27,19,29,31)(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);(1,3,5,15,6,13,18,14)(2,7,11,21,9,20,10,25)(4,12,17,29,16,27,30,28)(8,19,23,31,24,26,22,32);(1,4)(2,8)(3,12)(5,17)(6,16)(7,19)(9,24)(10,22)(11,23)(13,27)(14,28)(15,29)(18,30)(20,26)(21,31)(25,32)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3745);