Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #442 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| prime radical | C23×C53 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C22×C4×C53 ◁ C2×#97×C53 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 89 points:
(2,3)(5,6)(7,15)(8,20,23,36)(9,24)(10,27)(11,16)(12,18)(13,22,30,17)(14,26)(19,33)(21,32,25,35)(28,31,34,29)(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);(1,2,4,3)(5,7,9,18,11,19,26,10)(6,12,14,15,16,27,24,33)(8,17,25,34,23,22,21,28)(13,20,31,32,30,36,29,35);(1,3,4,2)(5,8,11,23)(6,13,16,30)(7,17,19,22)(9,25,26,21)(10,28,18,34)(12,20,27,36)(14,31,24,29)(15,32,33,35)2120 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6129);