Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#13197 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #121 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1954 |
| prime radical | C2×#135×C53 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C22×C8×C53 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,7,12,5,11,15,25,8,13,22,10,6,23,4,21)(3,14,18,30,19,9,16,26,20,27,24,32,17,28,31,29)(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);(1,3,15,31,6,17,7,24,8,20,4,16,5,19,22,18)(2,9,25,30,23,14,12,29,13,28,21,32,11,27,10,26);(1,4,6,22,8,15,5,7)(2,10,11,21,13,12,23,25)(3,16,17,18,20,31,19,24)(9,26,27,32,28,29,14,30)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6185);