Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C53 · Schur multiplier (H2) = C23 |
| direct product | C2×C53×#1328 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | #99 |
| commutator | C23 | C22×C8×C53 |
| Frattini, cosocle | C2×M16 | C23×C53 |
| socle | C22×C53 | #324 |
| prime radical | C24×C53 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C22×M16×C53 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 87 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 87 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);(56,57,59,64,60,65,70,78,61,63,69,75,71,76,82,85)(58,62,67,72,68,74,81,83,66,73,79,77,80,84,87,86);(56,58)(57,62,63,73)(59,67)(60,68)(61,66)(64,77,75,72)(65,74,76,84)(69,79)(70,81)(71,80)(78,86,85,83)(82,87)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 4445);