Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C24 |
| direct product | C53×#13045 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C2×C53 | #1889 |
| prime radical | C53×#3025 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G | |
654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,9,7,11,5,18)(3,12,15,28,16,29,19,27)(4,10,13,22,17,26,20,25)(8,21,23,31,24,30,32,14)(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);(1,3,13,21,7,16,20,30)(2,8,22,28,11,24,25,27)(4,14,6,15,17,31,5,19)(9,23,26,29,18,32,10,12);(2,9)(8,23)(10,25)(11,18)(12,27)(22,26)(24,32)(28,29)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6033);