Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#12646 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #102 |
| commutator | C8 | C2×C42×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | C4×#102 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,13,9,14,8,24)(3,15,17,25,20,28,19,32)(4,11,7,12,5,22,21,29)(10,18,26,23,27,31,30,16)(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);(1,3,7,18,8,19,4,16,9,20,21,31,6,17,5,23)(2,10,11,25,13,26,12,28,14,27,22,32,24,30,29,15);(1,4,6,7,9,5,8,21)(2,11,13,12,14,22,24,29)(3,16,17,18,20,23,19,31)(10,25,26,28,27,32,30,15)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5634);