Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C53 · Schur multiplier (H2) = C2×C4 |
| direct product | C53×#13152 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| socle | C22×C53 | C4×D8 |
| prime radical | C2×#135×C53 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C8×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)(3,4)(5,6,11,18,13,19,12,31)(7,20,8,26,25,35,24,32)(9,28,10,30,29,17,21,16)(14,23,33,27,15,34,36,22)(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);(5,7,9,22,11,8,10,23,13,25,29,27,12,24,21,34)(6,14,30,35,31,36,28,26,19,15,16,20,18,33,17,32);(2,4)(5,8,9,23,11,25,10,27,13,24,29,34,12,7,21,22)(6,15,16,35,31,33,17,26,19,14,30,20,18,36,28,32)2756 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6140);