Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C2×D4 |
| commutator | C2×C4 | C22×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C8×D4 |
| prime radical | C23×C53 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C22×C16×C53 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2)(3,4)(5,6,10,14,11,15,8,22,12,16,23,29,9,25,24,34)(7,13,19,26,20,27,17,30,21,28,31,35,18,33,32,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(2,4)(5,7,10,19,11,20,8,17,12,21,23,31,9,18,24,32)(6,13,14,26,15,27,22,30,16,28,29,35,25,33,34,36);(5,8,9,10,12,24,11,23)(6,14,15,22,16,29,25,34)(7,17,18,19,21,32,20,31)(13,26,27,30,28,35,33,36)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6149);