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×#13281 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C53 | C2×D8 |
| commutator | C2×C8 | C22×C4×C53 |
| Frattini, cosocle | C22×C8 | C23×C53 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C53 ◁ C2×C42×C53 ◁ C2×C4×C8×C53 ◁ G | |
366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 97 |
| transitive degree | μt(G) = 13568 |
A minimal faithful permutation representation — 3 generators on 97 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,31)(17,21)(18,34)(19,24)(20,28)(22,26)(23,32)(27,36)(29,44)(30,40)(33,38)(35,42)(37,43)(39,41)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97);(1,3,4,2)(13,15,26,41,18,29,33,43,19,30,16,27,32,42,17,28)(14,20,21,35,23,36,31,40,24,37,38,44,34,39,22,25);(5,7,8,12)(6,9,10,11)(13,16,18,17,19,26,32,33)(14,21,23,31,24,38,34,22)(15,27,29,28,30,41,42,43)(20,35,36,40,37,44,39,25)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 6269);