Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #127 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1955 |
| prime radical | C53×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,9,4,10,17,25,6,11,19,26,18,8,23,24)(3,12,16,27,15,7,20,22,14,13,28,29,30,21,32,31)(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);(1,2,5,9,4,10,17,25,6,11,19,26,18,8,23,24)(3,13,28,27,15,21,32,22,14,12,16,29,30,7,20,31);(1,3,4,15,6,14,18,30)(2,7,8,12,11,21,10,13)(5,16,17,20,19,28,23,32)(9,22,24,27,26,31,25,29)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6188);