Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#12656 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C53 | C2×D4 |
| commutator | C4 | C2×C4×C8×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C2×C4⋊C8 |
| prime radical | C23×C53 | C2×M16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C53 ◁ C2×C4×C8×C53 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(1,2,4,3)(5,6)(7,19,22,32)(8,24)(9,15)(10,16)(11,17)(12,18)(13,21,29,27)(14,26)(20,34,36,30)(23,33,35,28)(25,31)(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);(5,7,10,23,11,21,8,20,12,22,25,35,9,27,26,36)(6,13,16,30,17,19,24,33,18,29,31,34,15,32,14,28);(1,3,4,2)(5,8,9,10,12,26,11,25)(6,14,15,31,18,24,17,16)(7,20,27,23,22,36,21,35)(13,28,32,34,29,33,19,30)4664 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 5644);