Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#7163 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C53 | D4 |
| commutator | C4 | C4×C16×C53 |
| Frattini, cosocle | C2×C4×C8 | C22×C53 |
| socle | C22×C53 | #336 |
| prime radical | C23×C53 | C4⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,17,14,19,24,22,15,20,13,18,23,11,16,21)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(2,5,7,6)(9,11,13,22,14,10,16,18,15,19,12,21,23,20,24,17)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 151);