Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13129 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #124 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1924 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,11,7,12,22,28,5,10,19,23,20,26,32,13)(3,8,9,21,15,24,25,29,14,4,16,17,27,18,30,31)(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);(1,3,6,16,7,15,22,30,5,14,19,9,20,27,32,25)(2,8,23,21,12,24,13,29,10,4,11,17,26,18,28,31);(1,4)(2,9,10,16)(3,13,14,28)(5,8)(6,21)(7,18)(11,27,23,15)(12,25,26,30)(17,19)(20,24)(22,29)(31,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6117);