Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C53 · Schur multiplier (H2) = C25 |
| direct product | C2×C53×#2208 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C53 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C53 |
| prime radical | C22×D8×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 75 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 4 generators on 75 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75);(4,6)(7,8,11,15)(9,12,18,21)(10,19,17,14)(13,22,20,16);(3,4,5,6);(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20)3074 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 26562);