Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C53 · Schur multiplier (H2) = C1 |
| direct product | C53×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C53 | D4 |
| commutator | C4 | C2×C32×C53 |
| Frattini, cosocle | C2×C32 | C22×C53 |
| socle | C2×C53 | C4⋊C32 |
| prime radical | C22×C53 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C53 ◁ C2×C32×C53 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 2 generators on 117 points:
(1,2,5,12,6,13,23,35,7,10,24,30,27,31,49,52,8,11,25,33,28,34,50,55,3,15,16,38,17,39,41,57,9,14,26,36,29,37,51,56,18,32,42,53,44,54,60,63,4,20,21,46,22,47,48,62,19,40,43,58,45,59,61,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,3,4,7,9,19,8,18)(2,10,11,15,14,32,20,40)(5,16,21,24,26,43,25,42)(6,17,22,27,29,45,28,44)(12,30,33,38,36,53,46,58)(13,31,34,39,37,54,47,59)(23,41,48,49,51,61,50,60)(35,52,55,57,56,63,62,64)5936 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 532);