C3xDic18 - GroupNames

G = C₃xDic₁₈ order 216 = 2³·3³

Direct product of C₃ and Dic₁₈

direct product, metacyclic, supersoluble, monomial

Aliases: C₃xDic₁₈, C₃₆.₅C₆, C₁₂.₇D₉, C₆.₁₉D₁₈, Dic₉.₂C₆, C₃².₃Dic₆, C₄.(C₃xD₉), (C₃xC₉):₃Q₈, C₉:₃(C₃xQ₈), C₆.₆(S₃xC₆), C₂.₃(C₆xD₉), C₁₂.₁(C₃xS₃), (C₃xC₃₆).₂C₂, C₁₈.₉(C₂xC₆), (C₃xC₆).₄₅D₆, (C₃xC₁₂).₁₂S₃, C₃.₁(C₃xDic₆), (C₃xDic₉).₂C₂, (C₃xC₁₈).₁₃C₂², SmallGroup(216,43)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₈ — C₃xDic₁₈

Chief series

C₁ — C₃ — C₉ — C₁₈ — C₃xC₁₈ — C₃xDic₉ — C₃xDic₁₈

Lower central

C₉ — C₁₈ — C₃xDic₁₈

Upper central

C₁ — C₆ — C₁₂

Generators and relations for C₃xDic₁₈
G = < a,b,c | a³=b³⁶=1, c²=b¹⁸, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 108 in 42 conjugacy classes, 24 normal (20 characteristic)
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, D₆, C₂xC₆, D₉, C₃xS₃, Dic₆, C₃xQ₈, D₁₈, S₃xC₆, C₃xD₉, Dic₁₈, C₃xDic₆, C₆xD₉, C₃xDic₁₈

C₁

C₂

C₃

₂C₃

C₄

₉C₄

C₆

₂C₆

C₃²

C₉

₂C₉

₉Q₈

C₁₂

₂C₁₂

₃Dic₃

₉C₁₂

C₁₈

C₃xC₆

₂C₁₈

C₃xC₉

₃Dic₆

₉C₃xQ₈

C₃₆

Dic₉

C₃xC₁₂

₂C₃₆

₃C₃xDic₃

C₃xC₁₈

Dic₁₈

₃C₃xDic₆

C₃xC₃₆

C₃xDic₉

C₃xDic₁₈

Smallest permutation representation of C₃xDic₁₈
►On 72 points

Generators in S₇₂

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

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

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

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

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

C₃xDic₁₈ is a maximal subgroup of
C₆.D₃₆ C₃:Dic₃₆ D₁₂.D₉ C₁₂.D₁₈ Dic₁₈:S₃ D₁₂:D₉ Dic₉.D₆ C₃xQ₈xD₉

63 conjugacy classes

class	1	2	3A	3B	3C	3D	3E	4A	4B	4C	6A	6B	6C	6D	6E	9A	···	9I	12A	···	12H	12I	12J	12K	12L	18A	···	18I	36A	···	36R
order	1	2	3	3	3	3	3	4	4	4	6	6	6	6	6	9	···	9	12	···	12	12	12	12	12	18	···	18	36	···	36
size	1	1	1	1	2	2	2	2	18	18	1	1	2	2	2	2	···	2	2	···	2	18	18	18	18	2	···	2	2	···	2

63 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	2
type	+	+	+				+	-	+	+			-	+			-
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	Q₈	D₆	D₉	C₃xS₃	C₃xQ₈	Dic₆	D₁₈	S₃xC₆	C₃xD₉	Dic₁₈	C₃xDic₆	C₆xD₉	C₃xDic₁₈
kernel	C₃xDic₁₈	C₃xDic₉	C₃xC₃₆	Dic₁₈	Dic₉	C₃₆	C₃xC₁₂	C₃xC₉	C₃xC₆	C₁₂	C₁₂	C₉	C₃²	C₆	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	1	2	4	2	1	1	1	3	2	2	2	3	2	6	6	4	6	12

Matrix representation of C₃xDic₁₈ ►in GL₂(F₃₇) generated by

10	0
0	10

32	0
0	22

0	1
36	0

G:=sub<GL(2,GF(37))| [10,0,0,10],[32,0,0,22],[0,36,1,0] >;

C₃xDic₁₈ in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{18}

% in TeX

G:=Group("C3xDic18");

// GroupNames label

G:=SmallGroup(216,43);

// by ID

G=gap.SmallGroup(216,43);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,169,79,3604,208,5189]);

// Polycyclic

G:=Group<a,b,c|a^3=b^36=1,c^2=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

Export

Subgroup lattice of C₃xDic₁₈ in TeX

G = C3xDic18 order 216 = 23·33

Direct product of C3 and Dic18

G = C₃xDic₁₈ order 216 = 2³·3³

Direct product of C₃ and Dic₁₈