Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C53 · Schur multiplier (H2) = C25 |
| direct product | C4×C53×#401 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C53 | C23 |
| commutator | C23 | C23×C4×C53 |
| Frattini, cosocle | C24 | C24×C53 |
| socle, prime radical | C24×C53 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C53 ◁ G | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53)(54,55,56,57);(58,59,61,63)(60,62,64,65)(66,67,69,71)(68,70,72,73)(74,75,77,78)(76,80,81,79);(58,60,61,64)(59,62,63,65)(66,68)(67,70)(69,72)(71,73)(75,78)(76,81);(59,63)(62,65)(68,72)(70,73)(74,76,77,81)(75,79,78,80)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6854);