Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C53 · Schur multiplier (H2) = C22×C4 |
| direct product | C53×#12614 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C16×C53 | D4 |
| commutator | C4 | C2×C4×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C8×D4 |
| prime radical | C2×#41×C53 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16×C53 ◁ C22×C16×C53 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(5,6,9,15,10,14,22,29,11,16,23,30,8,12,21,25)(7,13,18,27,19,26,32,35,20,28,33,36,17,24,31,34)(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);(1,2,3,4)(6,12,16,14)(13,24,28,26)(15,25,30,29)(27,34,36,35);(5,7,9,18,10,19,22,32,11,20,23,33,8,17,21,31)(6,13,15,27,14,26,29,35,16,28,30,36,12,24,25,34)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5602);