Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C53 · Schur multiplier (H2) = C2 |
| direct product | C53×#7101 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | D8 |
| commutator | C8 | C2×C16×C53 |
| Frattini, cosocle | C82 | C22×C53 |
| socle, prime radical | C22×C53 | #335 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C4×C8×C53 ◁ C82×C53 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,22)(19,26)(20,27)(23,28)(24,32)(25,31)(29,30)(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);(17,18,23,32,20,24,19,22,21,25,26,30,27,29,28,31)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 89);