C9:Dic12 - GroupNames

G = C₉⋊Dic₁₂ order 432 = 2⁴·3³

The semidirect product of C₉ and Dic₁₂ acting via Dic₁₂/Dic₆=C₂

Aliases: C₉⋊₂Dic₁₂, C₃₆.₁₆D₆, C₁₈.₁₅D₁₂, C₁₂.₃₃D₁₈, Dic₆.₁D₉, C₉⋊C₈.S₃, C₁₂.₈S₃², (C₃×C₉)⋊₃Q₁₆, C₄.₄(S₃×D₉), (C₃×C₁₂).₈₀D₆, (C₃×C₁₈).₁₂D₄, C₃⋊₁(C₉⋊Q₁₆), C₆.₄(C₉⋊D₄), (C₃×Dic₆).₄S₃, (C₉×Dic₆).₂C₂, C₂.₇(C₉⋊D₁₂), C₁₂.D₉.₃C₂, (C₃×C₃₆).₁₅C₂², C₆.₁₇(C₃⋊D₁₂), C₃.₂(C₃²⋊₃Q₁₆), C₃².₃(C₃⋊Q₁₆), (C₃×C₉⋊C₈).₁C₂, (C₃×C₆).₄₈(C₃⋊D₄), SmallGroup(432,75)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₃₆ — C₉⋊Dic₁₂

Chief series

C₁ — C₃ — C₃² — C₃×C₉ — C₃×C₁₈ — C₃×C₃₆ — C₉×Dic₆ — C₉⋊Dic₁₂

Lower central

C₃×C₉ — C₃×C₁₈ — C₃×C₃₆ — C₉⋊Dic₁₂

Upper central

C₁ — C₂ — C₄

Generators and relations for C₉⋊Dic₁₂
G = < a,b,c | a⁹=b²⁴=1, c²=b¹², bab^-1=a^-1, ac=ca, cbc^-1=b^-1 >

Subgroups: 372 in 66 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, C₂₄, Dic₆, Dic₆, C₃×Q₈, C₃×C₉, Dic₉, C₃₆, C₃₆, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, Dic₁₂, C₃⋊Q₁₆, C₃×C₁₈, C₉⋊C₈, Dic₁₈, Q₈×C₉, C₃×C₃⋊C₈, C₃×Dic₆, C₃²⋊₄Q₈, C₉×Dic₃, C₉⋊Dic₃, C₃×C₃₆, C₉⋊Q₁₆, C₃²⋊₃Q₁₆, C₃×C₉⋊C₈, C₉×Dic₆, C₁₂.D₉, C₉⋊Dic₁₂
Quotients: C₁, C₂, C₂², S₃, D₄, D₆, Q₁₆, D₉, D₁₂, C₃⋊D₄, D₁₈, S₃², Dic₁₂, C₃⋊Q₁₆, C₉⋊D₄, C₃⋊D₁₂, S₃×D₉, C₉⋊Q₁₆, C₃²⋊₃Q₁₆, C₉⋊D₁₂, C₉⋊Dic₁₂

Smallest permutation representation of C₉⋊Dic₁₂
►On 144 points

Generators in S₁₄₄

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

(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)

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

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

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

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

51 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	18A	18B	18C	18D	18E	18F	24A	24B	24C	24D	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	18	18	18	18	18	18	24	24	24	24	36	···	36	36	···	36
size	1	1	2	2	4	2	12	108	2	2	4	18	18	2	2	2	4	4	4	2	2	4	4	4	12	12	2	2	2	4	4	4	18	18	18	18	4	···	4	12	···	12

51 irreducible representations

dim	1	1	1	1	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	S₃	S₃	D₄	D₆	D₆	Q₁₆	D₉	D₁₂	C₃⋊D₄	D₁₈	Dic₁₂	C₉⋊D₄	S₃²	C₃⋊Q₁₆	C₃⋊D₁₂	S₃×D₉	C₉⋊Q₁₆	C₃²⋊₃Q₁₆	C₉⋊D₁₂	C₉⋊Dic₁₂
kernel	C₉⋊Dic₁₂	C₃×C₉⋊C₈	C₉×Dic₆	C₁₂.D₉	C₉⋊C₈	C₃×Dic₆	C₃×C₁₈	C₃₆	C₃×C₁₂	C₃×C₉	Dic₆	C₁₈	C₃×C₆	C₁₂	C₉	C₆	C₁₂	C₃²	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	2	3	2	2	3	4	6	1	1	1	3	3	2	3	6

Matrix representation of C₉⋊Dic₁₂ ►in GL₄(𝔽₇₃) generated by

1	0	0	0
0	1	0	0
0	0	32	0
0	0	46	16

68	55	0	0
18	50	0	0
0	0	8	48
0	0	61	65

55	71	0	0
53	18	0	0
0	0	1	0
0	0	24	72

G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,32,46,0,0,0,16],[68,18,0,0,55,50,0,0,0,0,8,61,0,0,48,65],[55,53,0,0,71,18,0,0,0,0,1,24,0,0,0,72] >;

C₉⋊Dic₁₂ in GAP, Magma, Sage, TeX

C_9\rtimes {\rm Dic}_{12}

% in TeX

G:=Group("C9:Dic12");

// GroupNames label

G:=SmallGroup(432,75);

// by ID

G=gap.SmallGroup(432,75);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,3091,662,4037,7069]);

// Polycyclic

G:=Group<a,b,c|a^9=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;

// generators/relations

G = C9⋊Dic12 order 432 = 24·33

The semidirect product of C9 and Dic12 acting via Dic12/Dic6=C2

G = C₉⋊Dic₁₂ order 432 = 2⁴·3³

The semidirect product of C₉ and Dic₁₂ acting via Dic₁₂/Dic₆=C₂