Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#13108 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #387 |
| commutator | C22×C4 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C2×C53 | #1917 |
| prime radical | C23×C53 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C22×C8×C53 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,12,5,13,21,17,7,11,24,31,22,10,29,30)(3,4,9,20,15,18,27,32,16,19,28,23,14,8,25,26)(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);(1,3,6,9,5,15,21,27,7,16,24,28,22,14,29,25)(2,8,12,26,13,4,17,20,11,18,31,32,10,19,30,23);(1,4,5,18,7,19,22,8)(2,9,10,25,11,28,13,27)(3,12,14,30,16,31,15,17)(6,23,21,26,24,20,29,32)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6096);