Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#13479 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×C53 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×C4×C8×C53 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,3,4)(5,6)(7,20)(8,23)(9,17)(10,16)(11,22)(12,30)(13,19)(14,18)(15,26)(21,29)(24,31)(25,34)(27,35)(28,33)(32,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89);(5,7,10,19,12,22,11,30,13,16,20,6,14,17,9,18)(8,24,26,21,28,35,27,33,29,15,31,23,32,34,25,36);(2,4)(5,8,13,29)(6,15,19,24)(7,21,16,23)(9,26,11,31)(10,25,20,27)(12,32,14,28)(17,33,22,36)(18,35,30,34)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6467);