Dic3xC18 - GroupNames

G = Dic₃xC₁₈ order 216 = 2³·3³

Direct product of C₁₈ and Dic₃

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: Dic₃xC₁₈, C₆:C₃₆, C₁₈.₂₂D₆, C₆².₁₀C₆, (C₃xC₁₈):₁C₄, C₃:₂(C₂xC₃₆), C₂².(S₃xC₉), C₆.₃₂(S₃xC₆), C₂.₂(S₃xC₁₈), (C₂xC₆).₃C₁₈, (C₃xC₆).₇C₁₂, (C₆xC₁₈).₁C₂, (C₂xC₁₈).₆S₃, C₆.₄(C₂xC₁₈), (C₆xDic₃).C₃, C₃.₄(C₆xDic₃), C₆.₉(C₃xDic₃), C₃².₃(C₂xC₁₂), (C₃xDic₃).₆C₆, (C₃xC₁₈).₁₁C₂², (C₃xC₉):₇(C₂xC₄), (C₃xC₆).₂₁(C₂xC₆), (C₂xC₆).₂₀(C₃xS₃), (C₂xC₁₈)o(C₆xDic₃), (C₂xC₁₈)o(C₃xDic₃), SmallGroup(216,56)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — Dic₃xC₁₈

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₈ — C₉xDic₃ — Dic₃xC₁₈

Lower central

C₃ — Dic₃xC₁₈

Upper central

C₁ — C₂xC₁₈

Generators and relations for Dic₃xC₁₈
G = < a,b,c | a¹⁸=b⁶=1, c²=b³, ab=ba, ac=ca, cbc^-1=b^-1 >

Subgroups: 86 in 58 conjugacy classes, 39 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₆, C₆, C₆, C₂xC₄, C₉, C₉, C₃², Dic₃, C₁₂, C₂xC₆, C₂xC₆, C₁₈, C₁₈, C₁₈, C₃xC₆, C₃xC₆, C₂xDic₃, C₂xC₁₂, C₃xC₉, C₃₆, C₂xC₁₈, C₂xC₁₈, C₃xDic₃, C₆², C₃xC₁₈, C₃xC₁₈, C₂xC₃₆, C₆xDic₃, C₉xDic₃, C₆xC₁₈, Dic₃xC₁₈
Quotients: C₁, C₂, C₃, C₄, C₂², S₃, C₆, C₂xC₄, C₉, Dic₃, C₁₂, D₆, C₂xC₆, C₁₈, C₃xS₃, C₂xDic₃, C₂xC₁₂, C₃₆, C₂xC₁₈, C₃xDic₃, S₃xC₆, S₃xC₉, C₂xC₃₆, C₆xDic₃, C₉xDic₃, S₃xC₁₈, Dic₃xC₁₈

Smallest permutation representation of Dic₃xC₁₈
►On 72 points

Generators in S₇₂

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

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

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

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

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

Dic₃xC₁₈ is a maximal subgroup of Dic₉:Dic₃ C₁₈.Dic₆ Dic₃:Dic₉ D₁₈:Dic₃ C₆.₁₈D₃₆ D₁₈.₃D₆ S₃xC₂xC₃₆

108 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	···	6F	6G	···	6O	9A	···	9F	9G	···	9L	12A	···	12H	18A	···	18R	18S	···	18AJ	36A	···	36X
order	1	2	2	2	3	3	3	3	3	4	4	4	4	6	···	6	6	···	6	9	···	9	9	···	9	12	···	12	18	···	18	18	···	18	36	···	36
size	1	1	1	1	1	1	2	2	2	3	3	3	3	1	···	1	2	···	2	1	···	1	2	···	2	3	···	3	1	···	1	2	···	2	3	···	3

108 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+	+										+	-	+
image	C₁	C₂	C₂	C₃	C₄	C₆	C₆	C₉	C₁₂	C₁₈	C₁₈	C₃₆	S₃	Dic₃	D₆	C₃xS₃	C₃xDic₃	S₃xC₆	S₃xC₉	C₉xDic₃	S₃xC₁₈
kernel	Dic₃xC₁₈	C₉xDic₃	C₆xC₁₈	C₆xDic₃	C₃xC₁₈	C₃xDic₃	C₆²	C₂xDic₃	C₃xC₆	Dic₃	C₂xC₆	C₆	C₂xC₁₈	C₁₈	C₁₈	C₂xC₆	C₆	C₆	C₂²	C₂	C₂
# reps	1	2	1	2	4	4	2	6	8	12	6	24	1	2	1	2	4	2	6	12	6

Matrix representation of Dic₃xC₁₈ ►in GL₃(F₃₇) generated by

25	0	0
0	9	0
0	0	9

1	0	0
0	11	34
0	0	27

1	0	0
0	4	6
0	28	33

G:=sub<GL(3,GF(37))| [25,0,0,0,9,0,0,0,9],[1,0,0,0,11,0,0,34,27],[1,0,0,0,4,28,0,6,33] >;

Dic₃xC₁₈ in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{18}

% in TeX

G:=Group("Dic3xC18");

// GroupNames label

G:=SmallGroup(216,56);

// by ID

G=gap.SmallGroup(216,56);

# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,122,5189]);

// Polycyclic

G:=Group<a,b,c|a^18=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

G = Dic3xC18 order 216 = 23·33

Direct product of C18 and Dic3

G = Dic₃xC₁₈ order 216 = 2³·3³

Direct product of C₁₈ and Dic₃