Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13127 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #388 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1924 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C8×C53 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,5,10,4,19,7,11,21,30,23,29,22,9)(3,13,16,32,17,8,24,25,18,27,14,28,15,26,20,31)(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);(1,3,4,20,23,15,6,14,7,18,22,24,5,17,21,16)(2,8,19,32,29,13,12,31,11,26,9,28,10,27,30,25);(1,4,5,21,7,22,23,6)(2,9,10,12,11,19,29,30)(3,14,15,24,18,16,17,20)(8,25,13,32,26,31,27,28)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6115);