Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C53 · Schur multiplier (H2) = C22 |
| direct product | C53×#7472 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C53 | D16 |
| commutator | C16 | C42×C53 |
| Frattini, cosocle | C22×C16 | C22×C53 |
| socle | C22×C53 | #366 |
| prime radical | C23×C53 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C22×C8×C53 ◁ C22×C16×C53 ◁ G | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 6784 |
A minimal faithful permutation representation — 2 generators on 121 points:
(5,6,9,16,13,19,12,37)(7,20,22,50,26,51,25,61)(8,17,28,45,30,47,29,62)(10,15,31,43,34,44,33,46)(11,35,32,48,36,63,27,18)(14,38,39,54,42,53,64,21)(23,49,55,66,58,67,57,41)(24,59,56,65,60,68,52,40)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,2,3,4)(5,7,10,23,11,24,28,38,9,22,31,55,32,56,30,54,13,26,34,58,36,60,29,53,12,25,33,57,27,52,8,21)(6,14,17,40,18,41,46,61,37,64,62,68,63,67,44,51,19,42,47,65,48,66,43,50,16,39,45,59,35,49,15,20)4028 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 460);