C9xD9 - GroupNames

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(1 6 2 7 3 8 4 9 5)(10 14 18 13 17 12 16 11 15)

(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)

G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,6,2,7,3,8,4,9,5)(10,14,18,13,17,12,16,11,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)>;

G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,6,2,7,3,8,4,9,5)(10,14,18,13,17,12,16,11,15), (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,6,2,7,3,8,4,9,5),(10,14,18,13,17,12,16,11,15)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,10),(9,11)]])

G:=TransitiveGroup(18,74);

C₉xD₉ is a maximal subgroup of
C₂₇:₃C₁₈ C₉:C₅₄ C₉²:S₃ C₉².S₃ C₉²:₂S₃ C₉²:₇C₆ C₉²:₈C₆ C₉²:₃S₃ C₉²:₄S₃ C₉²:₆S₃ C₉²:₅S₃
C₉xD₉ is a maximal quotient of
C₉:S₃:C₉ C₂₇:₃C₁₈

54 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	6A	6B	9A	···	9F	9G	···	9AM	18A	···	18F
order	1	2	3	3	3	3	3	6	6	9	···	9	9	···	9	18	···	18
size	1	9	1	1	2	2	2	9	9	1	···	1	2	···	2	9	···	9

54 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2
type	+	+					+	+
image	C₁	C₂	C₃	C₆	C₉	C₁₈	S₃	D₉	C₃xS₃	C₃xD₉	S₃xC₉	C₉xD₉
kernel	C₉xD₉	C₉²	C₃xD₉	C₃xC₉	D₉	C₉	C₃xC₉	C₉	C₃²	C₃	C₃	C₁
# reps	1	1	2	2	6	6	1	3	2	6	6	18

Matrix representation of C₉xD₉ ►in GL₂(F₁₉) generated by

6	0
0	6

16	0
0	6

0	6
16	0

G:=sub<GL(2,GF(19))| [6,0,0,6],[16,0,0,6],[0,16,6,0] >;

C₉xD₉ in GAP, Magma, Sage, TeX

C_9\times D_9

% in TeX

G:=Group("C9xD9");

// GroupNames label

G:=SmallGroup(162,3);

// by ID

G=gap.SmallGroup(162,3);

# by ID

G:=PCGroup([5,-2,-3,-3,-3,-3,36,1803,138,2704]);

// Polycyclic

G:=Group<a,b,c|a^9=b^9=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;

// generators/relations

Export

Subgroup lattice of C₉xD₉ in TeX

G = C9xD9 order 162 = 2·34

Direct product of C9 and D9

G = C₉xD₉ order 162 = 2·3⁴

Direct product of C₉ and D₉