Dic18:S3 - GroupNames

G = Dic₁₈:S₃ order 432 = 2⁴·3³

3^rd semidirect product of Dic₁₈ and S₃ acting via S₃/C₃=C₂

Aliases: Dic₆:₄D₉, C₃₆.₂₄D₆, Dic₁₈:₃S₃, C₁₂.₂₂D₁₈, Dic₉.₁D₆, Dic₃.₄D₁₈, C₉:S₃:Q₈, C₉:₁(S₃xQ₈), C₃:₁(Q₈xD₉), C₁₂.₁₆S₃², C₄.₂₀(S₃xD₉), (C₉xDic₆):₆C₂, (C₃xC₁₂).₉₂D₆, C₁₈.D₆.C₂, C₆.₄(C₂²xD₉), (C₃xDic₁₈):₇C₂, C₉:Dic₆:₄C₂, C₃².₃(S₃xQ₈), C₁₈.₄(C₂²xS₃), (C₃xC₁₈).₄C₂³, (C₃xDic₃).₄D₆, (C₃xDic₆).₈S₃, (C₃xC₃₆).₂₇C₂², C₉:Dic₃.₈C₂², (C₃xDic₉).₁C₂², (C₉xDic₃).₄C₂², C₃.₁(Dic₃.D₆), (C₃xC₉):₂(C₂xQ₈), C₂.₈(C₂xS₃xD₉), C₆.₂₃(C₂xS₃²), (C₄xC₉:S₃).₁C₂, (C₂xC₉:S₃).₆C₂², (C₃xC₆).₇₂(C₂²xS₃), SmallGroup(432,283)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₈ — Dic₁₈:S₃

Chief series

C₁ — C₃ — C₃² — C₃xC₉ — C₃xC₁₈ — C₉xDic₃ — C₁₈.D₆ — Dic₁₈:S₃

Lower central

C₃xC₉ — C₃xC₁₈ — Dic₁₈:S₃

Upper central

C₁ — C₂ — C₄

Generators and relations for Dic₁₈:S₃
G = < a,b,c,d | a³⁶=c³=d²=1, b²=a¹⁸, bab^-1=a^-1, ac=ca, dad=a¹⁷, bc=cb, bd=db, dcd=c^-1 >

Subgroups: 784 in 130 conjugacy classes, 43 normal (27 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, Q₈, C₉, C₉, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₂xQ₈, D₉, C₁₈, C₁₈, C₃:S₃, C₃xC₆, Dic₆, Dic₆, C₄xS₃, C₃xQ₈, C₃xC₉, Dic₉, Dic₉, C₃₆, C₃₆, D₁₈, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, C₂xC₃:S₃, S₃xQ₈, C₉:S₃, C₃xC₁₈, Dic₁₈, Dic₁₈, C₄xD₉, Q₈xC₉, C₆.D₆, C₃²:₂Q₈, C₃xDic₆, C₃xDic₆, C₄xC₃:S₃, C₃xDic₉, C₉xDic₃, C₉:Dic₃, C₃xC₃₆, C₂xC₉:S₃, Q₈xD₉, Dic₃.D₆, C₉:Dic₆, C₁₈.D₆, C₃xDic₁₈, C₉xDic₆, C₄xC₉:S₃, Dic₁₈:S₃
Quotients: C₁, C₂, C₂², S₃, Q₈, C₂³, D₆, C₂xQ₈, D₉, C₂²xS₃, D₁₈, S₃², S₃xQ₈, C₂²xD₉, C₂xS₃², S₃xD₉, Q₈xD₉, Dic₃.D₆, C₂xS₃xD₉, Dic₁₈:S₃

Smallest permutation representation of Dic₁₈:S₃
►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 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)

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

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

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

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

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

51 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	4E	4F	6A	6B	6C	9A	9B	9C	9D	9E	9F	12A	12B	12C	12D	12E	12F	12G	12H	18A	18B	18C	18D	18E	18F	36A	···	36I	36J	···	36O
order	1	2	2	2	3	3	3	4	4	4	4	4	4	6	6	6	9	9	9	9	9	9	12	12	12	12	12	12	12	12	18	18	18	18	18	18	36	···	36	36	···	36
size	1	1	27	27	2	2	4	2	6	6	18	18	54	2	2	4	2	2	2	4	4	4	4	4	4	4	12	12	36	36	2	2	2	4	4	4	4	···	4	12	···	12

51 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	-	+	+	+	+	+	+	+	+	-	-	+	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	S₃	Q₈	D₆	D₆	D₆	D₆	D₉	D₁₈	D₁₈	S₃²	S₃xQ₈	S₃xQ₈	C₂xS₃²	S₃xD₉	Q₈xD₉	Dic₃.D₆	C₂xS₃xD₉	Dic₁₈:S₃
kernel	Dic₁₈:S₃	C₉:Dic₆	C₁₈.D₆	C₃xDic₁₈	C₉xDic₆	C₄xC₉:S₃	Dic₁₈	C₃xDic₆	C₉:S₃	Dic₉	C₃₆	C₃xDic₃	C₃xC₁₂	Dic₆	Dic₃	C₁₂	C₁₂	C₉	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	2	1	1	1	1	1	2	2	1	2	1	3	6	3	1	1	1	1	3	3	2	3	6

Matrix representation of Dic₁₈:S₃ ►in GL₆(F₃₇)

0	1	0	0	0	0
36	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	6	20
0	0	0	0	17	26

3	8	0	0	0	0
8	34	0	0	0	0
0	0	36	0	0	0
0	0	0	36	0	0
0	0	0	0	0	1
0	0	0	0	1	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	36	1	0	0
0	0	36	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(37))| [0,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,17,0,0,0,0,20,26],[3,8,0,0,0,0,8,34,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

Dic₁₈:S₃ in GAP, Magma, Sage, TeX

{\rm Dic}_{18}\rtimes S_3

% in TeX

G:=Group("Dic18:S3");

// GroupNames label

G:=SmallGroup(432,283);

// by ID

G=gap.SmallGroup(432,283);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,64,254,135,58,3091,662,4037,7069]);

// Polycyclic

G:=Group<a,b,c,d|a^36=c^3=d^2=1,b^2=a^18,b*a*b^-1=a^-1,a*c=c*a,d*a*d=a^17,b*c=c*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = Dic18:S3 order 432 = 24·33

3rd semidirect product of Dic18 and S3 acting via S3/C3=C2

G = Dic₁₈:S₃ order 432 = 2⁴·3³

3^rd semidirect product of Dic₁₈ and S₃ acting via S₃/C₃=C₂