Dic3.D18 - GroupNames

G = Dic₃.D₁₈ order 432 = 2⁴·3³

5^th non-split extension by Dic₃ of D₁₈ acting via D₁₈/D₉=C₂

Aliases: D₆.₃D₁₈, C₆².₆₆D₆, Dic₉.₁₂D₆, Dic₃.₅D₁₈, C₃:D₄:₃D₉, (S₃xC₆).₅D₆, (C₂xC₁₈).₄D₆, C₉:D₁₂:₄C₂, C₉:₄(C₄oD₁₂), (C₆xDic₉):₇C₂, (S₃xDic₉):₃C₂, (C₂xDic₉):₃S₃, (C₂xC₆).₂₀D₁₈, C₃:₃(D₄:₂D₉), C₂².₁(S₃xD₉), C₉:Dic₆:₅C₂, C₁₈.D₆:₂C₂, (C₃xDic₃).₅D₆, C₆.₂₁(C₂²xD₉), C₆.D₁₈:₂C₂, (S₃xC₁₈).₃C₂², C₁₈.₂₁(C₂²xS₃), (C₃xC₁₈).₂₁C₂³, (C₆xC₁₈).₁₅C₂², C₉:Dic₃.₆C₂², C₃.₂(D₆.₃D₆), (C₉xDic₃).₅C₂², C₃².₄(D₄:₂S₃), (C₃xDic₉).₁₂C₂², (C₂xC₆).₄S₃², C₆.₄₀(C₂xS₃²), C₂.₂₁(C₂xS₃xD₉), (C₃xC₉):₉(C₄oD₄), (C₉xC₃:D₄):₁C₂, (C₃xC₃:D₄).₁S₃, (C₂xC₉:S₃).₄C₂², (C₃xC₆).₈₉(C₂²xS₃), SmallGroup(432,309)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₈ — Dic₃.D₁₈

Chief series

C₁ — C₃ — C₃² — C₃xC₉ — C₃xC₁₈ — S₃xC₁₈ — S₃xDic₉ — Dic₃.D₁₈

Lower central

C₃xC₉ — C₃xC₁₈ — Dic₃.D₁₈

Upper central

C₁ — C₂ — C₂²

Generators and relations for Dic₃.D₁₈
G = < a,b,c,d | a⁶=c¹⁸=1, b²=d²=a³, bab^-1=cac^-1=dad^-1=a^-1, cbc^-1=dbd^-1=a³b, dcd^-1=c^-1 >

Subgroups: 828 in 136 conjugacy classes, 41 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, C₉, C₉, C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₄oD₄, D₉, C₁₈, C₁₈, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xC₉, Dic₉, Dic₉, C₃₆, D₁₈, C₂xC₁₈, C₂xC₁₈, C₃xDic₃, C₃xDic₃, C₃:Dic₃, S₃xC₆, C₂xC₃:S₃, C₆², C₄oD₁₂, D₄:₂S₃, S₃xC₉, C₉:S₃, C₃xC₁₈, C₃xC₁₈, Dic₁₈, C₄xD₉, C₂xDic₉, C₂xDic₉, C₉:D₄, D₄xC₉, S₃xDic₃, C₆.D₆, C₃:D₁₂, C₃²:₂Q₈, C₆xDic₃, C₃xC₃:D₄, C₃²:₇D₄, C₃xDic₉, C₉xDic₃, C₉:Dic₃, S₃xC₁₈, C₂xC₉:S₃, C₆xC₁₈, D₄:₂D₉, D₆.₃D₆, C₉:Dic₆, C₁₈.D₆, S₃xDic₉, C₉:D₁₂, C₆xDic₉, C₉xC₃:D₄, C₆.D₁₈, Dic₃.D₁₈
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄oD₄, D₉, C₂²xS₃, D₁₈, S₃², C₄oD₁₂, D₄:₂S₃, C₂²xD₉, C₂xS₃², S₃xD₉, D₄:₂D₉, D₆.₃D₆, C₂xS₃xD₉, Dic₃.D₁₈

Smallest permutation representation of Dic₃.D₁₈
►On 72 points

Generators in S₇₂

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

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

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	+		+	-	+	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₆	D₆	D₆	D₆	D₆	C₄oD₄	D₉	D₁₈	D₁₈	D₁₈	C₄oD₁₂	S₃²	D₄:₂S₃	C₂xS₃²	S₃xD₉	D₄:₂D₉	D₆.₃D₆	C₂xS₃xD₉	Dic₃.D₁₈
kernel	Dic₃.D₁₈	C₉:Dic₆	C₁₈.D₆	S₃xDic₉	C₉:D₁₂	C₆xDic₉	C₉xC₃:D₄	C₆.D₁₈	C₂xDic₉	C₃xC₃:D₄	Dic₉	C₂xC₁₈	C₃xDic₃	S₃xC₆	C₆²	C₃xC₉	C₃:D₄	Dic₃	D₆	C₂xC₆	C₉	C₂xC₆	C₃²	C₆	C₂²	C₃	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	2	1	1	1	1	2	3	3	3	3	4	1	1	1	3	3	2	3	6

Matrix representation of Dic₃.D₁₈ ►in GL₄(F₃₇) generated by

27	0	0	0
33	11	0	0
0	0	1	0
0	0	0	1

29	5	0	0
24	8	0	0
0	0	1	0
0	0	0	1

29	5	0	0
17	8	0	0
0	0	12	0
0	0	0	34

26	30	0	0
28	11	0	0
0	0	0	34
0	0	12	0

G:=sub<GL(4,GF(37))| [27,33,0,0,0,11,0,0,0,0,1,0,0,0,0,1],[29,24,0,0,5,8,0,0,0,0,1,0,0,0,0,1],[29,17,0,0,5,8,0,0,0,0,12,0,0,0,0,34],[26,28,0,0,30,11,0,0,0,0,0,12,0,0,34,0] >;

Dic₃.D₁₈ in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_{18}

% in TeX

G:=Group("Dic3.D18");

// GroupNames label

G:=SmallGroup(432,309);

// by ID

G=gap.SmallGroup(432,309);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,3091,662,4037,7069]);

// Polycyclic

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

// generators/relations

G = Dic3.D18 order 432 = 24·33

5th non-split extension by Dic3 of D18 acting via D18/D9=C2

G = Dic₃.D₁₈ order 432 = 2⁴·3³

5^th non-split extension by Dic₃ of D₁₈ acting via D₁₈/D₉=C₂