C12.D18 - GroupNames

G = C₁₂.D₁₈ order 432 = 2⁴·3³

16^th non-split extension by C₁₂ of D₁₈ acting via D₁₈/C₉=C₂²

Aliases: C₃₆.₁₅D₆, C₁₂.₁₆D₁₈, Dic₆.₂D₉, Dic₁₈.₂S₃, C₁₂.₇S₃², (C₃×C₉)⋊₂Q₁₆, C₄.₁₈(S₃×D₉), C₉⋊₂(C₃⋊Q₁₆), (C₃×C₁₈).₁₁D₄, (C₃×C₁₂).₇₉D₆, C₃⋊₂(C₉⋊Q₁₆), C₆.₁₁(C₉⋊D₄), (C₉×Dic₆).₁C₂, (C₃×Dic₆).₃S₃, C₂.₇(D₆⋊D₉), C₁₈.₁₀(C₃⋊D₄), C₃₆.S₃.₁C₂, (C₃×C₃₆).₁₄C₂², (C₃×Dic₁₈).₃C₂, C₆.₁₅(D₆⋊S₃), C₃².₂(C₃⋊Q₁₆), C₃.₂(C₃²⋊₂Q₁₆), (C₃×C₆).₄₇(C₃⋊D₄), SmallGroup(432,74)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₃₆ — C₁₂.D₁₈

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃×C₁₈ — C₃×C₃₆ — C₉×Dic₆ — C₁₂.D₁₈

Lower central

C₃×C₉ — C₃×C₁₈ — C₃×C₃₆ — C₁₂.D₁₈

Upper central

C₁ — C₂ — C₄

Generators and relations for C₁₂.D₁₈
G = < a,b,c | a¹²=1, b¹⁸=a⁶, c²=a⁹, bab^-1=a^-1, cac^-1=a⁵, cbc^-1=a⁹b¹⁷ >

Subgroups: 272 in 62 conjugacy classes, 25 normal (all characteristic)
C₁, C₂, C₃, C₃, C₄, C₄, C₆, C₆, C₈, Q₈, C₉, C₉, C₃², Dic₃, C₁₂, C₁₂, Q₁₆, C₁₈, C₁₈, C₃×C₆, C₃⋊C₈, Dic₆, Dic₆, C₃×Q₈, C₃×C₉, Dic₉, C₃₆, C₃₆, C₃×Dic₃, C₃×C₁₂, C₃⋊Q₁₆, C₃×C₁₈, C₉⋊C₈, Dic₁₈, Q₈×C₉, C₃²⋊₄C₈, C₃×Dic₆, C₃×Dic₆, C₃×Dic₉, C₉×Dic₃, C₃×C₃₆, C₉⋊Q₁₆, C₃²⋊₂Q₁₆, C₃₆.S₃, C₃×Dic₁₈, C₉×Dic₆, C₁₂.D₁₈
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, Q₁₆, D₉, C₃⋊D₄, D₁₈, S₃², C₃⋊Q₁₆, C₉⋊D₄, D₆⋊S₃, S₃×D₉, C₉⋊Q₁₆, C₃²⋊₂Q₁₆, D₆⋊D₉, C₁₂.D₁₈

Smallest permutation representation of C₁₂.D₁₈
►On 144 points

Generators in S₁₄₄

(1 127 7 133 13 139 19 109 25 115 31 121)(2 122 32 116 26 110 20 140 14 134 8 128)(3 129 9 135 15 141 21 111 27 117 33 123)(4 124 34 118 28 112 22 142 16 136 10 130)(5 131 11 137 17 143 23 113 29 119 35 125)(6 126 36 120 30 114 24 144 18 138 12 132)(37 95 43 101 49 107 55 77 61 83 67 89)(38 90 68 84 62 78 56 108 50 102 44 96)(39 97 45 103 51 73 57 79 63 85 69 91)(40 92 70 86 64 80 58 74 52 104 46 98)(41 99 47 105 53 75 59 81 65 87 71 93)(42 94 72 88 66 82 60 76 54 106 48 100)

(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)(73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)

(1 91 115 63 19 73 133 45)(2 62 134 90 20 44 116 108)(3 89 117 61 21 107 135 43)(4 60 136 88 22 42 118 106)(5 87 119 59 23 105 137 41)(6 58 138 86 24 40 120 104)(7 85 121 57 25 103 139 39)(8 56 140 84 26 38 122 102)(9 83 123 55 27 101 141 37)(10 54 142 82 28 72 124 100)(11 81 125 53 29 99 143 71)(12 52 144 80 30 70 126 98)(13 79 127 51 31 97 109 69)(14 50 110 78 32 68 128 96)(15 77 129 49 33 95 111 67)(16 48 112 76 34 66 130 94)(17 75 131 47 35 93 113 65)(18 46 114 74 36 64 132 92)

G:=sub<Sym(144)| (1,127,7,133,13,139,19,109,25,115,31,121)(2,122,32,116,26,110,20,140,14,134,8,128)(3,129,9,135,15,141,21,111,27,117,33,123)(4,124,34,118,28,112,22,142,16,136,10,130)(5,131,11,137,17,143,23,113,29,119,35,125)(6,126,36,120,30,114,24,144,18,138,12,132)(37,95,43,101,49,107,55,77,61,83,67,89)(38,90,68,84,62,78,56,108,50,102,44,96)(39,97,45,103,51,73,57,79,63,85,69,91)(40,92,70,86,64,80,58,74,52,104,46,98)(41,99,47,105,53,75,59,81,65,87,71,93)(42,94,72,88,66,82,60,76,54,106,48,100), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,91,115,63,19,73,133,45)(2,62,134,90,20,44,116,108)(3,89,117,61,21,107,135,43)(4,60,136,88,22,42,118,106)(5,87,119,59,23,105,137,41)(6,58,138,86,24,40,120,104)(7,85,121,57,25,103,139,39)(8,56,140,84,26,38,122,102)(9,83,123,55,27,101,141,37)(10,54,142,82,28,72,124,100)(11,81,125,53,29,99,143,71)(12,52,144,80,30,70,126,98)(13,79,127,51,31,97,109,69)(14,50,110,78,32,68,128,96)(15,77,129,49,33,95,111,67)(16,48,112,76,34,66,130,94)(17,75,131,47,35,93,113,65)(18,46,114,74,36,64,132,92)>;

G:=Group( (1,127,7,133,13,139,19,109,25,115,31,121)(2,122,32,116,26,110,20,140,14,134,8,128)(3,129,9,135,15,141,21,111,27,117,33,123)(4,124,34,118,28,112,22,142,16,136,10,130)(5,131,11,137,17,143,23,113,29,119,35,125)(6,126,36,120,30,114,24,144,18,138,12,132)(37,95,43,101,49,107,55,77,61,83,67,89)(38,90,68,84,62,78,56,108,50,102,44,96)(39,97,45,103,51,73,57,79,63,85,69,91)(40,92,70,86,64,80,58,74,52,104,46,98)(41,99,47,105,53,75,59,81,65,87,71,93)(42,94,72,88,66,82,60,76,54,106,48,100), (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)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,91,115,63,19,73,133,45)(2,62,134,90,20,44,116,108)(3,89,117,61,21,107,135,43)(4,60,136,88,22,42,118,106)(5,87,119,59,23,105,137,41)(6,58,138,86,24,40,120,104)(7,85,121,57,25,103,139,39)(8,56,140,84,26,38,122,102)(9,83,123,55,27,101,141,37)(10,54,142,82,28,72,124,100)(11,81,125,53,29,99,143,71)(12,52,144,80,30,70,126,98)(13,79,127,51,31,97,109,69)(14,50,110,78,32,68,128,96)(15,77,129,49,33,95,111,67)(16,48,112,76,34,66,130,94)(17,75,131,47,35,93,113,65)(18,46,114,74,36,64,132,92) );

G=PermutationGroup([[(1,127,7,133,13,139,19,109,25,115,31,121),(2,122,32,116,26,110,20,140,14,134,8,128),(3,129,9,135,15,141,21,111,27,117,33,123),(4,124,34,118,28,112,22,142,16,136,10,130),(5,131,11,137,17,143,23,113,29,119,35,125),(6,126,36,120,30,114,24,144,18,138,12,132),(37,95,43,101,49,107,55,77,61,83,67,89),(38,90,68,84,62,78,56,108,50,102,44,96),(39,97,45,103,51,73,57,79,63,85,69,91),(40,92,70,86,64,80,58,74,52,104,46,98),(41,99,47,105,53,75,59,81,65,87,71,93),(42,94,72,88,66,82,60,76,54,106,48,100)], [(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),(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,91,115,63,19,73,133,45),(2,62,134,90,20,44,116,108),(3,89,117,61,21,107,135,43),(4,60,136,88,22,42,118,106),(5,87,119,59,23,105,137,41),(6,58,138,86,24,40,120,104),(7,85,121,57,25,103,139,39),(8,56,140,84,26,38,122,102),(9,83,123,55,27,101,141,37),(10,54,142,82,28,72,124,100),(11,81,125,53,29,99,143,71),(12,52,144,80,30,70,126,98),(13,79,127,51,31,97,109,69),(14,50,110,78,32,68,128,96),(15,77,129,49,33,95,111,67),(16,48,112,76,34,66,130,94),(17,75,131,47,35,93,113,65),(18,46,114,74,36,64,132,92)]])

48 conjugacy classes

class	1	2	3A	3B	3C	4A	4B	4C	6A	6B	6C	8A	8B	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	3	3	3	4	4	4	6	6	6	8	8	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	2	2	4	2	12	36	2	2	4	54	54	2	2	2	4	4	4	4	4	4	4	12	12	36	36	2	2	2	4	4	4	4	···	4	12	···	12

48 irreducible representations

dim	1	1	1	1	2	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	S₃	S₃	D₄	D₆	D₆	Q₁₆	D₉	C₃⋊D₄	C₃⋊D₄	D₁₈	C₉⋊D₄	S₃²	C₃⋊Q₁₆	C₃⋊Q₁₆	D₆⋊S₃	S₃×D₉	C₉⋊Q₁₆	C₃²⋊₂Q₁₆	D₆⋊D₉	C₁₂.D₁₈
kernel	C₁₂.D₁₈	C₃₆.S₃	C₃×Dic₁₈	C₉×Dic₆	Dic₁₈	C₃×Dic₆	C₃×C₁₈	C₃₆	C₃×C₁₂	C₃×C₉	Dic₆	C₁₈	C₃×C₆	C₁₂	C₆	C₁₂	C₉	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	2	3	2	2	3	6	1	1	1	1	3	3	2	3	6

Matrix representation of C₁₂.D₁₈ ►in GL₆(𝔽₇₃)

28	71	0	0	0	0
64	45	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

4	17	0	0	0	0
72	69	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	20	71
0	0	0	0	2	18

26	41	0	0	0	0
2	6	0	0	0	0
0	0	0	72	0	0
0	0	72	0	0	0
0	0	0	0	26	39
0	0	0	0	65	47

G:=sub<GL(6,GF(73))| [28,64,0,0,0,0,71,45,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[4,72,0,0,0,0,17,69,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,20,2,0,0,0,0,71,18],[26,2,0,0,0,0,41,6,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,26,65,0,0,0,0,39,47] >;

C₁₂.D₁₈ in GAP, Magma, Sage, TeX

C_{12}.D_{18}

% in TeX

G:=Group("C12.D18");

// GroupNames label

G:=SmallGroup(432,74);

// by ID

G=gap.SmallGroup(432,74);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,64,254,135,58,3091,662,4037,7069]);

// Polycyclic

G:=Group<a,b,c|a^12=1,b^18=a^6,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^9*b^17>;

// generators/relations

G = C12.D18 order 432 = 24·33

16th non-split extension by C12 of D18 acting via D18/C9=C22

G = C₁₂.D₁₈ order 432 = 2⁴·3³

16^th non-split extension by C₁₂ of D₁₈ acting via D₁₈/C₉=C₂²