Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13182 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #459 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1947 |
| 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:
(1,2,7,12,8,13,6,24)(3,14,18,30,19,31,17,29)(4,20,21,28,22,11,5,10)(9,23,25,15,27,32,26,16)(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);(1,3,7,18,8,19,6,17)(2,9,12,25,13,27,24,26)(4,15,21,32,22,16,5,23)(10,14,20,30,28,31,11,29);(1,4,6,5,8,22,7,21)(2,10,12,20,13,28,24,11)(3,15,17,23,19,16,18,32)(9,14,25,30,27,31,26,29)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6170);