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×#12624 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | C2×#99 |
| commutator | C23 | C2×C42×C53 |
| Frattini, cosocle | C2×#41 | C23×C53 |
| socle | C2×C53 | C4×#99 |
| prime radical | C53×#525 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C2×C8×C53 ◁ C2×#131×C53 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,5,4,7,10,19,17)(3,11,14,24,13,8,21,12)(6,9,18,16,20,23,30,29)(15,22,28,26,27,25,31,32)(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);(1,3,6,15)(2,8,9,22)(4,12,16,26)(5,14,18,28)(7,13,20,27)(10,11,23,25)(17,24,29,32)(19,21,30,31);(1,4,5,10,7,17,19,2)(3,12,14,11,13,24,21,8)(6,16,18,23,20,29,30,9)(15,26,28,25,27,32,31,22)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5612);