C3xC18.D4 - GroupNames

G = C₃xC₁₈.D₄ order 432 = 2⁴·3³

Direct product of C₃ and C₁₈.D₄

direct product, metabelian, supersoluble, monomial

Aliases: C₃xC₁₈.D₄, C₆².₁₂₂D₆, C₆².₁₇Dic₃, (C₆xC₁₈):₄C₄, (C₂xC₆):₃Dic₉, (C₂xC₁₈):₁₀C₁₂, (C₆xDic₉):₄C₂, (C₂xDic₉):₈C₆, C₁₈.₂₇(C₃xD₄), (C₃xC₁₈).₄₄D₄, (C₂xC₆).₅₀D₁₈, C₂.₅(C₆xDic₉), C₂³.₃(C₃xD₉), C₂².₇(C₆xD₉), (C₂²xC₆).₅D₉, C₁₈.₂₃(C₂xC₁₂), C₆.₃₅(C₉:D₄), (C₂xC₆²).₂₁S₃, C₆.₁₈(C₆xDic₃), C₆.₂₁(C₂xDic₉), C₂²:₃(C₃xDic₉), (C₂²xC₁₈).₁₄C₆, (C₆xC₁₈).₃₆C₂², C₃².₃(C₆.D₄), (C₂xC₆xC₁₈).₄C₂, C₉:₅(C₃xC₂²:C₄), C₂.₃(C₃xC₉:D₄), (C₃xC₉):₈(C₂²:C₄), (C₂xC₆).₄₆(S₃xC₆), C₆.₃₀(C₃xC₃:D₄), (C₃xC₁₈).₃₅(C₂xC₄), (C₂xC₁₈).₂₇(C₂xC₆), (C₃xC₆).₉₆(C₃:D₄), (C₂²xC₆).₂₄(C₃xS₃), (C₃xC₆).₅₈(C₂xDic₃), (C₂xC₆).₁₈(C₃xDic₃), C₃.₁(C₃xC₆.D₄), SmallGroup(432,164)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₈ — C₃xC₁₈.D₄

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₂xC₁₈ — C₆xC₁₈ — C₆xDic₉ — C₃xC₁₈.D₄

Lower central

C₉ — C₁₈ — C₃xC₁₈.D₄

Upper central

C₁ — C₂xC₆ — C₂²xC₆

Generators and relations for C₃xC₁₈.D₄
G = < a,b,c,d | a³=b¹⁸=c⁴=1, d²=b⁹, ab=ba, ac=ca, ad=da, cbc^-1=dbd^-1=b^-1, dcd^-1=b⁹c^-1 >

Subgroups: 342 in 134 conjugacy classes, 54 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₃, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂xC₄, C₂³, C₉, C₉, C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₂xC₆, C₂²:C₄, C₁₈, C₁₈, C₁₈, C₃xC₆, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₂²xC₆, C₂²xC₆, C₃xC₉, Dic₉, C₂xC₁₈, C₂xC₁₈, C₂xC₁₈, C₃xDic₃, C₆², C₆², C₆², C₆.D₄, C₃xC₂²:C₄, C₃xC₁₈, C₃xC₁₈, C₃xC₁₈, C₂xDic₉, C₂²xC₁₈, C₂²xC₁₈, C₆xDic₃, C₂xC₆², C₃xDic₉, C₆xC₁₈, C₆xC₁₈, C₆xC₁₈, C₁₈.D₄, C₃xC₆.D₄, C₆xDic₉, C₂xC₆xC₁₈, C₃xC₁₈.D₄
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, D₄, Dic₃, C₁₂, D₆, C₂xC₆, C₂²:C₄, D₉, C₃xS₃, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, Dic₉, D₁₈, C₃xDic₃, S₃xC₆, C₆.D₄, C₃xC₂²:C₄, C₃xD₉, C₂xDic₉, C₉:D₄, C₆xDic₃, C₃xC₃:D₄, C₃xDic₉, C₆xD₉, C₁₈.D₄, C₃xC₆.D₄, C₆xDic₉, C₃xC₉:D₄, C₃xC₁₈.D₄

Smallest permutation representation of C₃xC₁₈.D₄
►On 72 points

Generators in S₇₂

(1 13 7)(2 14 8)(3 15 9)(4 16 10)(5 17 11)(6 18 12)(19 25 31)(20 26 32)(21 27 33)(22 28 34)(23 29 35)(24 30 36)(37 49 43)(38 50 44)(39 51 45)(40 52 46)(41 53 47)(42 54 48)(55 61 67)(56 62 68)(57 63 69)(58 64 70)(59 65 71)(60 66 72)

(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)(37 38 39 40 41 42 43 44 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)

(1 21 53 64)(2 20 54 63)(3 19 37 62)(4 36 38 61)(5 35 39 60)(6 34 40 59)(7 33 41 58)(8 32 42 57)(9 31 43 56)(10 30 44 55)(11 29 45 72)(12 28 46 71)(13 27 47 70)(14 26 48 69)(15 25 49 68)(16 24 50 67)(17 23 51 66)(18 22 52 65)

(1 55 10 64)(2 72 11 63)(3 71 12 62)(4 70 13 61)(5 69 14 60)(6 68 15 59)(7 67 16 58)(8 66 17 57)(9 65 18 56)(19 37 28 46)(20 54 29 45)(21 53 30 44)(22 52 31 43)(23 51 32 42)(24 50 33 41)(25 49 34 40)(26 48 35 39)(27 47 36 38)

G:=sub<Sym(72)| (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38)>;

G:=Group( (1,13,7)(2,14,8)(3,15,9)(4,16,10)(5,17,11)(6,18,12)(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,29,35)(24,30,36)(37,49,43)(38,50,44)(39,51,45)(40,52,46)(41,53,47)(42,54,48)(55,61,67)(56,62,68)(57,63,69)(58,64,70)(59,65,71)(60,66,72), (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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72), (1,21,53,64)(2,20,54,63)(3,19,37,62)(4,36,38,61)(5,35,39,60)(6,34,40,59)(7,33,41,58)(8,32,42,57)(9,31,43,56)(10,30,44,55)(11,29,45,72)(12,28,46,71)(13,27,47,70)(14,26,48,69)(15,25,49,68)(16,24,50,67)(17,23,51,66)(18,22,52,65), (1,55,10,64)(2,72,11,63)(3,71,12,62)(4,70,13,61)(5,69,14,60)(6,68,15,59)(7,67,16,58)(8,66,17,57)(9,65,18,56)(19,37,28,46)(20,54,29,45)(21,53,30,44)(22,52,31,43)(23,51,32,42)(24,50,33,41)(25,49,34,40)(26,48,35,39)(27,47,36,38) );

G=PermutationGroup([[(1,13,7),(2,14,8),(3,15,9),(4,16,10),(5,17,11),(6,18,12),(19,25,31),(20,26,32),(21,27,33),(22,28,34),(23,29,35),(24,30,36),(37,49,43),(38,50,44),(39,51,45),(40,52,46),(41,53,47),(42,54,48),(55,61,67),(56,62,68),(57,63,69),(58,64,70),(59,65,71),(60,66,72)], [(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),(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72)], [(1,21,53,64),(2,20,54,63),(3,19,37,62),(4,36,38,61),(5,35,39,60),(6,34,40,59),(7,33,41,58),(8,32,42,57),(9,31,43,56),(10,30,44,55),(11,29,45,72),(12,28,46,71),(13,27,47,70),(14,26,48,69),(15,25,49,68),(16,24,50,67),(17,23,51,66),(18,22,52,65)], [(1,55,10,64),(2,72,11,63),(3,71,12,62),(4,70,13,61),(5,69,14,60),(6,68,15,59),(7,67,16,58),(8,66,17,57),(9,65,18,56),(19,37,28,46),(20,54,29,45),(21,53,30,44),(22,52,31,43),(23,51,32,42),(24,50,33,41),(25,49,34,40),(26,48,35,39),(27,47,36,38)]])

126 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	···	6F	6G	···	6AE	9A	···	9I	12A	···	12H	18A	···	18BK
order	1	2	2	2	2	2	3	3	3	3	3	4	4	4	4	6	···	6	6	···	6	9	···	9	12	···	12	18	···	18
size	1	1	1	1	2	2	1	1	2	2	2	18	18	18	18	1	···	1	2	···	2	2	···	2	18	···	18	2	···	2

126 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+						+	+	-	+	+				-	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₁₂	S₃	D₄	Dic₃	D₆	D₉	C₃xS₃	C₃xD₄	C₃:D₄	Dic₉	D₁₈	C₃xDic₃	S₃xC₆	C₃xD₉	C₉:D₄	C₃xC₃:D₄	C₃xDic₉	C₆xD₉	C₃xC₉:D₄
kernel	C₃xC₁₈.D₄	C₆xDic₉	C₂xC₆xC₁₈	C₁₈.D₄	C₆xC₁₈	C₂xDic₉	C₂²xC₁₈	C₂xC₁₈	C₂xC₆²	C₃xC₁₈	C₆²	C₆²	C₂²xC₆	C₂²xC₆	C₁₈	C₃xC₆	C₂xC₆	C₂xC₆	C₂xC₆	C₂xC₆	C₂³	C₆	C₆	C₂²	C₂²	C₂
# reps	1	2	1	2	4	4	2	8	1	2	2	1	3	2	4	4	6	3	4	2	6	12	8	12	6	24

Matrix representation of C₃xC₁₈.D₄ ►in GL₄(F₃₇) generated by

1	0	0	0
0	1	0	0
0	0	26	0
0	0	0	26

10	0	0	0
5	26	0	0
0	0	28	0
0	0	0	4

31	3	0	0
0	6	0	0
0	0	0	1
0	0	36	0

31	3	0	0
13	6	0	0
0	0	0	1
0	0	36	0

G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,0,0,0,0,26],[10,5,0,0,0,26,0,0,0,0,28,0,0,0,0,4],[31,0,0,0,3,6,0,0,0,0,0,36,0,0,1,0],[31,13,0,0,3,6,0,0,0,0,0,36,0,0,1,0] >;

C₃xC₁₈.D₄ in GAP, Magma, Sage, TeX

C_3\times C_{18}.D_4

% in TeX

G:=Group("C3xC18.D4");

// GroupNames label

G:=SmallGroup(432,164);

// by ID

G=gap.SmallGroup(432,164);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,10085,292,14118]);

// Polycyclic

G:=Group<a,b,c,d|a^3=b^18=c^4=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^-1>;

// generators/relations

G = C3xC18.D4 order 432 = 24·33

Direct product of C3 and C18.D4

G = C₃xC₁₈.D₄ order 432 = 2⁴·3³

Direct product of C₃ and C₁₈.D₄