Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C16×C53 · Schur multiplier (H2) = C2×C8 |
| direct product | C53×#11410 |
| subgroup | G/H | |
|---|---|---|
| center | C82×C53 | C22 |
| commutator | C2 | C2×C4×C16×C53 |
| Frattini, cosocle | C4×C8 | C23×C53 |
| socle | C22×C53 | C2×C4×C8 |
| prime radical | C23×C53 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C82×C53 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 3 generators on 85 points:
(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32)(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);(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(22,28)(26,31)(29,32)8480 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4398);