Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C53 · Schur multiplier (H2) = C1 |
| direct product | C53×#7333 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C53 | #322 |
| commutator | C8 | C4×C8×C53 |
| Frattini, cosocle | #346 | C22×C53 |
| socle | C2×C53 | #1313 |
| prime radical | C22×C53 | #347 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C53 ◁ C53×#346 ◁ 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:
(1,2,3,8,4,10,14,24,5,9,15,26,17,21,31,32,6,11,13,25,16,28,30,22,18,27,12,23,29,7,19,20)(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);(2,7,9,10,11,21,27,28)(3,12,13,15)(4,16)(8,22,23,32,25,24,26,20)(14,31,30,19)(17,29)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 321);