C2xC9:C18 - GroupNames

G = C₂xC₉:C₁₈ order 324 = 2²·3⁴

Direct product of C₂ and C₉:C₁₈

direct product, metacyclic, supersoluble, monomial

Aliases: C₂xC₉:C₁₈, C₁₈:C₁₈, D₉:C₁₈, D₁₈:C₉, C₉:(C₂xC₁₈), C₉:C₉:C₂², (C₃xD₉).C₆, (C₆xD₉).C₃, C₆.₆(S₃xC₉), (C₃xC₉).₁D₆, C₆.₅(C₉:C₆), C₃.₃(S₃xC₁₈), (C₃xC₁₈).₅S₃, (C₃xC₁₈).₅C₆, C₃².₁₆(S₃xC₆), (C₂xC₉:C₉):C₂, (C₃xC₉).(C₂xC₆), C₃.₂(C₂xC₉:C₆), (C₃xC₆).₃₂(C₃xS₃), SmallGroup(324,64)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₉ — C₂xC₉:C₁₈

Chief series

C₁ — C₃ — C₉ — C₃xC₉ — C₉:C₉ — C₉:C₁₈ — C₂xC₉:C₁₈

Lower central

C₉ — C₂xC₉:C₁₈

Upper central

C₁ — C₆

Generators and relations for C₂xC₉:C₁₈
G = < a,b,c | a²=b⁹=c¹⁸=1, ab=ba, ac=ca, cbc^-1=b² >

Subgroups: 163 in 53 conjugacy classes, 25 normal (19 characteristic)
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, C₉, D₆, C₂xC₆, C₁₈, C₃xS₃, C₂xC₁₈, S₃xC₆, S₃xC₉, C₉:C₆, S₃xC₁₈, C₂xC₉:C₆, C₉:C₁₈, C₂xC₉:C₁₈

C₁

C₂

₉C₂

C₃

₂C₃

₉C₂²

C₆

₂C₆

₃S₃

₉C₆

C₉

C₃²

₂C₉

₃C₉

₆C₉

₃D₆

₉C₂xC₆

C₃xC₆

C₁₈

D₉

₂C₁₈

₃C₁₈

₃C₃xS₃

₆C₁₈

₉C₁₈

C₃xC₉

₂C₃xC₉

D₁₈

₃S₃xC₆

₉C₂xC₁₈

C₃xC₁₈

C₃xD₉

C₃xC₁₈

₂C₃xC₁₈

₃S₃xC₉

C₉:C₉

C₆xD₉

₃S₃xC₁₈

C₉:C₁₈

C₂xC₉:C₉

C₂xC₉:C₁₈

Smallest permutation representation of C₂xC₉:C₁₈
►On 36 points

Generators in S₃₆

(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)

(1 17 3 13 11 15 7 5 9)(2 4 12 8 10 18 14 16 6)(19 35 21 31 29 33 25 23 27)(20 22 30 26 28 36 32 34 24)

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36)

G:=sub<Sym(36)| (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,17,3,13,11,15,7,5,9)(2,4,12,8,10,18,14,16,6)(19,35,21,31,29,33,25,23,27)(20,22,30,26,28,36,32,34,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24), (1,17,3,13,11,15,7,5,9)(2,4,12,8,10,18,14,16,6)(19,35,21,31,29,33,25,23,27)(20,22,30,26,28,36,32,34,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36) );

G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24)], [(1,17,3,13,11,15,7,5,9),(2,4,12,8,10,18,14,16,6),(19,35,21,31,29,33,25,23,27),(20,22,30,26,28,36,32,34,24)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36)]])

60 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	6A	6B	6C	6D	6E	6F	6G	6H	6I	9A	···	9F	9G	···	9O	18A	···	18F	18G	···	18O	18P	···	18AA
order	1	2	2	2	3	3	3	3	3	6	6	6	6	6	6	6	6	6	9	···	9	9	···	9	18	···	18	18	···	18	18	···	18
size	1	1	9	9	1	1	2	2	2	1	1	2	2	2	9	9	9	9	3	···	3	6	···	6	3	···	3	6	···	6	9	···	9

60 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	6	6	6	6
type	+	+	+							+	+					+	+
image	C₁	C₂	C₂	C₃	C₆	C₆	C₉	C₁₈	C₁₈	S₃	D₆	C₃xS₃	S₃xC₆	S₃xC₉	S₃xC₁₈	C₉:C₆	C₂xC₉:C₆	C₉:C₁₈	C₂xC₉:C₁₈
kernel	C₂xC₉:C₁₈	C₉:C₁₈	C₂xC₉:C₉	C₆xD₉	C₃xD₉	C₃xC₁₈	D₁₈	D₉	C₁₈	C₃xC₁₈	C₃xC₉	C₃xC₆	C₃²	C₆	C₃	C₆	C₃	C₂	C₁
# reps	1	2	1	2	4	2	6	12	6	1	1	2	2	6	6	1	1	2	2

Matrix representation of C₂xC₉:C₁₈ ►in GL₈(F₁₉)

18	0	0	0	0	0	0	0
0	18	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

11	0	0	0	0	0	0	0
11	7	0	0	0	0	0	0
0	0	0	7	0	0	0	0
0	0	0	0	7	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	1
0	0	0	0	0	11	0	0
0	0	0	0	0	0	11	0

8	4	0	0	0	0	0	0
0	11	0	0	0	0	0	0
0	0	0	0	0	0	7	0
0	0	0	0	0	0	0	11
0	0	0	0	0	11	0	0
0	0	0	7	0	0	0	0
0	0	0	0	11	0	0	0
0	0	11	0	0	0	0	0

G:=sub<GL(8,GF(19))| [18,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,11,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,1,0,0],[8,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0,0,11,0,0,0,0,0,7,0,0,0,0,0,0,0,0,11,0,0,0,0] >;

C₂xC₉:C₁₈ in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes C_{18}

% in TeX

G:=Group("C2xC9:C18");

// GroupNames label

G:=SmallGroup(324,64);

// by ID

G=gap.SmallGroup(324,64);

# by ID

G:=PCGroup([6,-2,-2,-3,-3,-3,-3,68,5404,1096,208,7781]);

// Polycyclic

G:=Group<a,b,c|a^2=b^9=c^18=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;

// generators/relations

Export

Subgroup lattice of C₂xC₉:C₁₈ in TeX

18	0	0	0	0	0	0	0
0	18	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

11	0	0	0	0	0	0	0
11	7	0	0	0	0	0	0
0	0	0	7	0	0	0	0
0	0	0	0	7	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	1
0	0	0	0	0	11	0	0
0	0	0	0	0	0	11	0

8	4	0	0	0	0	0	0
0	11	0	0	0	0	0	0
0	0	0	0	0	0	7	0
0	0	0	0	0	0	0	11
0	0	0	0	0	11	0	0
0	0	0	7	0	0	0	0
0	0	0	0	11	0	0	0
0	0	11	0	0	0	0	0

18	0	0	0	0	0	0	0
0	18	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

11	0	0	0	0	0	0	0
11	7	0	0	0	0	0	0
0	0	0	7	0	0	0	0
0	0	0	0	7	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	1
0	0	0	0	0	11	0	0
0	0	0	0	0	0	11	0

8	4	0	0	0	0	0	0
0	11	0	0	0	0	0	0
0	0	0	0	0	0	7	0
0	0	0	0	0	0	0	11
0	0	0	0	0	11	0	0
0	0	0	7	0	0	0	0
0	0	0	0	11	0	0	0
0	0	11	0	0	0	0	0

G = C2xC9:C18 order 324 = 22·34

Direct product of C2 and C9:C18

G = C₂xC₉:C₁₈ order 324 = 2²·3⁴

Direct product of C₂ and C₉:C₁₈

18	0	0	0	0	0	0	0
0	18	0	0	0	0	0	0
0	0	1	0	0	0	0	0
0	0	0	1	0	0	0	0
0	0	0	0	1	0	0	0
0	0	0	0	0	1	0	0
0	0	0	0	0	0	1	0
0	0	0	0	0	0	0	1

11	0	0	0	0	0	0	0
11	7	0	0	0	0	0	0
0	0	0	7	0	0	0	0
0	0	0	0	7	0	0	0
0	0	1	0	0	0	0	0
0	0	0	0	0	0	0	1
0	0	0	0	0	11	0	0
0	0	0	0	0	0	11	0

8	4	0	0	0	0	0	0
0	11	0	0	0	0	0	0
0	0	0	0	0	0	7	0
0	0	0	0	0	0	0	11
0	0	0	0	0	11	0	0
0	0	0	7	0	0	0	0
0	0	0	0	11	0	0	0
0	0	11	0	0	0	0	0