D6.D18 - GroupNames

G = D₆.D₁₈ order 432 = 2⁴·3³

1^st non-split extension by D₆ of D₁₈ acting via D₁₈/C₁₈=C₂

Aliases: D₆.₅D₁₈, D₁₈.₇D₆, C₃₆.₄₃D₆, C₁₂.₄₃D₁₈, Dic₉.₈D₆, Dic₃.₈D₁₈, C₁₂.₆₅S₃², (C₄xS₃):₄D₉, (C₄xD₉):₄S₃, (S₃xC₃₆):₂C₂, (C₁₂xD₉):₇C₂, C₄.₂₈(S₃xD₉), C₉:D₁₂:₇C₂, D₆:D₉:₇C₂, C₉:₂(C₄oD₁₂), C₃:D₃₆:₇C₂, (S₃xC₁₂).₂S₃, (S₃xC₆).₂₆D₆, C₆.₈(C₂²xD₉), C₉:Dic₆:₇C₂, (C₃xC₁₂).₁₆₁D₆, C₁₈.₈(C₂²xS₃), (C₃xC₁₈).₈C₂³, C₃:₂(D₃₆:₅C₂), (C₆xD₉).₈C₂², (S₃xC₁₈).₉C₂², (C₃xC₃₆).₄₂C₂², (C₃xDic₃).₂₉D₆, C₃.₁(D₆.D₆), C₃².₂(C₄oD₁₂), C₉:Dic₃.₁₀C₂², (C₉xDic₃).₈C₂², (C₃xDic₉).₁₀C₂², (C₄xC₉:S₃):₇C₂, C₆.₂₇(C₂xS₃²), C₂.₁₂(C₂xS₃xD₉), (C₃xC₉):₅(C₄oD₄), (C₂xC₉:S₃).₈C₂², (C₃xC₆).₇₆(C₂²xS₃), SmallGroup(432,287)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₈ — D₆.D₁₈

Chief series

C₁ — C₃ — C₃² — C₃xC₉ — C₃xC₁₈ — S₃xC₁₈ — D₆:D₉ — D₆.D₁₈

Lower central

C₃xC₉ — C₃xC₁₈ — D₆.D₁₈

Upper central

C₁ — C₄

Generators and relations for D₆.D₁₈
G = < a,b,c,d | a¹⁸=b²=1, c⁶=d²=a⁹, bab=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a⁹b, dcd^-1=c⁵ >

Subgroups: 860 in 136 conjugacy classes, 41 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, C₉, C₉, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, C₂xC₆, C₄oD₄, D₉, C₁₈, C₁₈, C₃xS₃, C₃:S₃, C₃xC₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₃xC₉, Dic₉, Dic₉, C₃₆, C₃₆, D₁₈, D₁₈, C₂xC₁₈, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₄oD₁₂, C₃xD₉, S₃xC₉, C₉:S₃, C₃xC₁₈, Dic₁₈, C₄xD₉, C₄xD₉, D₃₆, C₉:D₄, C₂xC₃₆, D₆:S₃, C₃:D₁₂, C₃²:₂Q₈, S₃xC₁₂, S₃xC₁₂, C₄xC₃:S₃, C₃xDic₉, C₉xDic₃, C₉:Dic₃, C₃xC₃₆, C₆xD₉, S₃xC₁₈, C₂xC₉:S₃, D₃₆:₅C₂, D₆.D₆, C₉:Dic₆, C₃:D₃₆, D₆:D₉, C₉:D₁₂, C₁₂xD₉, S₃xC₃₆, C₄xC₉:S₃, D₆.D₁₈
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄oD₄, D₉, C₂²xS₃, D₁₈, S₃², C₄oD₁₂, C₂²xD₉, C₂xS₃², S₃xD₉, D₃₆:₅C₂, D₆.D₆, C₂xS₃xD₉, D₆.D₁₈

Smallest permutation representation of D₆.D₁₈
►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 44)(2 43)(3 42)(4 41)(5 40)(6 39)(7 38)(8 37)(9 54)(10 53)(11 52)(12 51)(13 50)(14 49)(15 48)(16 47)(17 46)(18 45)(19 63)(20 62)(21 61)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)

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

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

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

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

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

66 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	6F	6G	9A	9B	9C	9D	9E	9F	12A	12B	12C	12D	12E	12F	12G	12H	12I	12J	18A	18B	18C	18D	18E	18F	18G	···	18L	36A	···	36F	36G	···	36L	36M	···	36R
order	1	2	2	2	2	3	3	3	4	4	4	4	4	6	6	6	6	6	6	6	9	9	9	9	9	9	12	12	12	12	12	12	12	12	12	12	18	18	18	18	18	18	18	···	18	36	···	36	36	···	36	36	···	36
size	1	1	6	18	54	2	2	4	1	1	6	18	54	2	2	4	6	6	18	18	2	2	2	4	4	4	2	2	2	2	4	4	6	6	18	18	2	2	2	4	4	4	6	···	6	2	···	2	4	···	4	6	···	6

66 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	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	+				+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₆	D₆	D₆	D₆	D₆	D₆	C₄oD₄	D₉	D₁₈	D₁₈	D₁₈	C₄oD₁₂	C₄oD₁₂	D₃₆:₅C₂	S₃²	C₂xS₃²	S₃xD₉	D₆.D₆	C₂xS₃xD₉	D₆.D₁₈
kernel	D₆.D₁₈	C₉:Dic₆	C₃:D₃₆	D₆:D₉	C₉:D₁₂	C₁₂xD₉	S₃xC₃₆	C₄xC₉:S₃	C₄xD₉	S₃xC₁₂	Dic₉	C₃₆	D₁₈	C₃xDic₃	C₃xC₁₂	S₃xC₆	C₃xC₉	C₄xS₃	Dic₃	C₁₂	D₆	C₉	C₃²	C₃	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	3	3	3	3	4	4	12	1	1	3	2	3	6

Matrix representation of D₆.D₁₈ ►in GL₆(F₃₇)

36	0	0	0	0	0
0	36	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	17	11
0	0	0	0	26	6

20	2	0	0	0	0
4	17	0	0	0	0
0	0	36	0	0	0
0	0	0	36	0	0
0	0	0	0	20	26
0	0	0	0	6	17

6	0	0	0	0	0
0	6	0	0	0	0
0	0	0	36	0	0
0	0	1	1	0	0
0	0	0	0	36	0
0	0	0	0	0	36

20	2	0	0	0	0
3	17	0	0	0	0
0	0	0	36	0	0
0	0	36	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,26,0,0,0,0,11,6],[20,4,0,0,0,0,2,17,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,20,6,0,0,0,0,26,17],[6,0,0,0,0,0,0,6,0,0,0,0,0,0,0,1,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[20,3,0,0,0,0,2,17,0,0,0,0,0,0,0,36,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D₆.D₁₈ in GAP, Magma, Sage, TeX

D_6.D_{18}

% in TeX

G:=Group("D6.D18");

// GroupNames label

G:=SmallGroup(432,287);

// by ID

G=gap.SmallGroup(432,287);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,3091,662,4037,7069]);

// Polycyclic

G:=Group<a,b,c,d|a^18=b^2=1,c^6=d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^9*b,d*c*d^-1=c^5>;

// generators/relations

G = D6.D18 order 432 = 24·33

1st non-split extension by D6 of D18 acting via D18/C18=C2

G = D₆.D₁₈ order 432 = 2⁴·3³

1^st non-split extension by D₆ of D₁₈ acting via D₁₈/C₁₈=C₂