Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13188 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #476 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1948 |
| prime radical | C23×C53 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,13,8,14,7,12)(3,15,18,27,19,29,17,32)(4,11,5,10,21,25,20,26)(9,22,23,16,24,30,28,31)(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)(2,9,14,24)(4,16)(5,22)(6,17)(7,18)(8,19)(10,15,26,29)(11,27,25,32)(12,23,13,28)(20,30)(21,31);(1,4,6,5,8,21,7,20)(2,10,12,11,14,26,13,25)(3,16,17,22,19,31,18,30)(9,15,23,27,24,29,28,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6176);