Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13135 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #459 |
| 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×C4×C8×C53 ◁ G | |
324 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)(7,17)(8,20)(9,24)(10,12)(11,16)(13,18)(14,25)(15,19)(21,34)(22,33)(23,28)(26,31)(27,29)(30,35)(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);(1,3,4,2)(5,7,9,10,11,19,25,27)(6,12,14,17,16,29,24,15)(8,18,21,26,23,36,22,35)(13,28,30,34,32,20,31,33);(5,8,9,21,11,23,25,22)(6,13,14,30,16,32,24,31)(7,18,10,26,19,36,27,35)(12,28,17,34,29,20,15,33)2438 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6123);