Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C53 · Schur multiplier (H2) = C1 |
| direct product | C53×#7456 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C53 | #106 |
| commutator | C8 | C2×C16×C53 |
| Frattini, cosocle | C4×C16 | C22×C53 |
| socle | C2×C53 | #1367 |
| prime radical | C22×C53 | #364 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C53 ◁ C2×C16×C53 ◁ C4×C16×C53 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1696 |
A minimal faithful permutation representation — 2 generators on 85 points:
(1,2,5,12,7,14,3,16,8,13,18,29,23,30,19,31,9,15,25,10,22,11,20,28,6,26,24,27,4,21,17,32)(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);(1,3,4,5,6,17,22,24,9,20,23,25,8,19,7,18)(2,10,11,16,13,27,21,31,15,12,14,28,26,29,30,32)3392 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 444);