direct product, metabelian, supersoluble, monomial
Aliases: C2×C9⋊Dic6, C6⋊1Dic18, C18⋊1Dic6, Dic9.9D6, C62.60D6, Dic3.9D18, (C3×C18)⋊Q8, C9⋊2(C2×Dic6), C3⋊2(C2×Dic18), (C2×C18).15D6, (C2×C6).15D18, (C6×C18).9C22, (C2×Dic3).4D9, (C6×Dic9).7C2, (C6×Dic3).9S3, (C2×Dic9).4S3, (C3×C6).15Dic6, C6.15(C22×D9), C22.11(S3×D9), (C3×C18).15C23, C18.15(C22×S3), (Dic3×C18).6C2, (C3×Dic3).31D6, C6.4(C32⋊2Q8), C32.3(C2×Dic6), C9⋊Dic3.11C22, (C9×Dic3).9C22, (C3×Dic9).11C22, (C3×C9)⋊4(C2×Q8), C6.34(C2×S32), (C2×C6).21S32, C2.18(C2×S3×D9), (C2×C9⋊Dic3).7C2, C3.1(C2×C32⋊2Q8), (C3×C6).83(C22×S3), SmallGroup(432,303)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C9⋊Dic6
G = < a,b,c,d | a2=b9=c12=1, d2=c6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 652 in 130 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, C2×Dic6, C3×C18, C3×C18, Dic18, C2×Dic9, C2×Dic9, C2×C36, C32⋊2Q8, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, C2×Dic18, C2×C32⋊2Q8, C9⋊Dic6, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, C2×C9⋊Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, S32, C2×Dic6, Dic18, C22×D9, C32⋊2Q8, C2×S32, S3×D9, C2×Dic18, C2×C32⋊2Q8, C9⋊Dic6, C2×S3×D9, C2×C9⋊Dic6
►On 144 points
Generators in S144
(1 83)(2 84)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 119)(14 120)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 108)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 130)(38 131)(39 132)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 85)(59 86)(60 87)(61 142)(62 143)(63 144)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)
(1 142 110 9 138 118 5 134 114)(2 115 135 6 119 139 10 111 143)(3 144 112 11 140 120 7 136 116)(4 117 137 8 109 141 12 113 133)(13 70 80 17 62 84 21 66 76)(14 77 67 22 73 63 18 81 71)(15 72 82 19 64 74 23 68 78)(16 79 69 24 75 65 20 83 61)(25 59 132 33 55 128 29 51 124)(26 125 52 30 129 56 34 121 60)(27 49 122 35 57 130 31 53 126)(28 127 54 32 131 58 36 123 50)(37 102 92 45 98 88 41 106 96)(38 85 107 42 89 99 46 93 103)(39 104 94 47 100 90 43 108 86)(40 87 97 44 91 101 48 95 105)
(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)
(1 131 7 125)(2 130 8 124)(3 129 9 123)(4 128 10 122)(5 127 11 121)(6 126 12 132)(13 98 19 104)(14 97 20 103)(15 108 21 102)(16 107 22 101)(17 106 23 100)(18 105 24 99)(25 115 31 109)(26 114 32 120)(27 113 33 119)(28 112 34 118)(29 111 35 117)(30 110 36 116)(37 78 43 84)(38 77 44 83)(39 76 45 82)(40 75 46 81)(41 74 47 80)(42 73 48 79)(49 133 55 139)(50 144 56 138)(51 143 57 137)(52 142 58 136)(53 141 59 135)(54 140 60 134)(61 85 67 91)(62 96 68 90)(63 95 69 89)(64 94 70 88)(65 93 71 87)(66 92 72 86)
G:=sub<Sym(144)| (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,119)(14,120)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,108)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,130)(38,131)(39,132)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,85)(59,86)(60,87)(61,142)(62,143)(63,144)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141), (1,142,110,9,138,118,5,134,114)(2,115,135,6,119,139,10,111,143)(3,144,112,11,140,120,7,136,116)(4,117,137,8,109,141,12,113,133)(13,70,80,17,62,84,21,66,76)(14,77,67,22,73,63,18,81,71)(15,72,82,19,64,74,23,68,78)(16,79,69,24,75,65,20,83,61)(25,59,132,33,55,128,29,51,124)(26,125,52,30,129,56,34,121,60)(27,49,122,35,57,130,31,53,126)(28,127,54,32,131,58,36,123,50)(37,102,92,45,98,88,41,106,96)(38,85,107,42,89,99,46,93,103)(39,104,94,47,100,90,43,108,86)(40,87,97,44,91,101,48,95,105), (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), (1,131,7,125)(2,130,8,124)(3,129,9,123)(4,128,10,122)(5,127,11,121)(6,126,12,132)(13,98,19,104)(14,97,20,103)(15,108,21,102)(16,107,22,101)(17,106,23,100)(18,105,24,99)(25,115,31,109)(26,114,32,120)(27,113,33,119)(28,112,34,118)(29,111,35,117)(30,110,36,116)(37,78,43,84)(38,77,44,83)(39,76,45,82)(40,75,46,81)(41,74,47,80)(42,73,48,79)(49,133,55,139)(50,144,56,138)(51,143,57,137)(52,142,58,136)(53,141,59,135)(54,140,60,134)(61,85,67,91)(62,96,68,90)(63,95,69,89)(64,94,70,88)(65,93,71,87)(66,92,72,86)>;
G:=Group( (1,83)(2,84)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,119)(14,120)(15,109)(16,110)(17,111)(18,112)(19,113)(20,114)(21,115)(22,116)(23,117)(24,118)(25,108)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,130)(38,131)(39,132)(40,121)(41,122)(42,123)(43,124)(44,125)(45,126)(46,127)(47,128)(48,129)(49,88)(50,89)(51,90)(52,91)(53,92)(54,93)(55,94)(56,95)(57,96)(58,85)(59,86)(60,87)(61,142)(62,143)(63,144)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141), (1,142,110,9,138,118,5,134,114)(2,115,135,6,119,139,10,111,143)(3,144,112,11,140,120,7,136,116)(4,117,137,8,109,141,12,113,133)(13,70,80,17,62,84,21,66,76)(14,77,67,22,73,63,18,81,71)(15,72,82,19,64,74,23,68,78)(16,79,69,24,75,65,20,83,61)(25,59,132,33,55,128,29,51,124)(26,125,52,30,129,56,34,121,60)(27,49,122,35,57,130,31,53,126)(28,127,54,32,131,58,36,123,50)(37,102,92,45,98,88,41,106,96)(38,85,107,42,89,99,46,93,103)(39,104,94,47,100,90,43,108,86)(40,87,97,44,91,101,48,95,105), (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), (1,131,7,125)(2,130,8,124)(3,129,9,123)(4,128,10,122)(5,127,11,121)(6,126,12,132)(13,98,19,104)(14,97,20,103)(15,108,21,102)(16,107,22,101)(17,106,23,100)(18,105,24,99)(25,115,31,109)(26,114,32,120)(27,113,33,119)(28,112,34,118)(29,111,35,117)(30,110,36,116)(37,78,43,84)(38,77,44,83)(39,76,45,82)(40,75,46,81)(41,74,47,80)(42,73,48,79)(49,133,55,139)(50,144,56,138)(51,143,57,137)(52,142,58,136)(53,141,59,135)(54,140,60,134)(61,85,67,91)(62,96,68,90)(63,95,69,89)(64,94,70,88)(65,93,71,87)(66,92,72,86) );
G=PermutationGroup([[(1,83),(2,84),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,119),(14,120),(15,109),(16,110),(17,111),(18,112),(19,113),(20,114),(21,115),(22,116),(23,117),(24,118),(25,108),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,130),(38,131),(39,132),(40,121),(41,122),(42,123),(43,124),(44,125),(45,126),(46,127),(47,128),(48,129),(49,88),(50,89),(51,90),(52,91),(53,92),(54,93),(55,94),(56,95),(57,96),(58,85),(59,86),(60,87),(61,142),(62,143),(63,144),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141)], [(1,142,110,9,138,118,5,134,114),(2,115,135,6,119,139,10,111,143),(3,144,112,11,140,120,7,136,116),(4,117,137,8,109,141,12,113,133),(13,70,80,17,62,84,21,66,76),(14,77,67,22,73,63,18,81,71),(15,72,82,19,64,74,23,68,78),(16,79,69,24,75,65,20,83,61),(25,59,132,33,55,128,29,51,124),(26,125,52,30,129,56,34,121,60),(27,49,122,35,57,130,31,53,126),(28,127,54,32,131,58,36,123,50),(37,102,92,45,98,88,41,106,96),(38,85,107,42,89,99,46,93,103),(39,104,94,47,100,90,43,108,86),(40,87,97,44,91,101,48,95,105)], [(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)], [(1,131,7,125),(2,130,8,124),(3,129,9,123),(4,128,10,122),(5,127,11,121),(6,126,12,132),(13,98,19,104),(14,97,20,103),(15,108,21,102),(16,107,22,101),(17,106,23,100),(18,105,24,99),(25,115,31,109),(26,114,32,120),(27,113,33,119),(28,112,34,118),(29,111,35,117),(30,110,36,116),(37,78,43,84),(38,77,44,83),(39,76,45,82),(40,75,46,81),(41,74,47,80),(42,73,48,79),(49,133,55,139),(50,144,56,138),(51,143,57,137),(52,142,58,136),(53,141,59,135),(54,140,60,134),(61,85,67,91),(62,96,68,90),(63,95,69,89),(64,94,70,88),(65,93,71,87),(66,92,72,86)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | 6H | 6I | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 18A | ··· | 18I | 18J | ··· | 18R | 36A | ··· | 36L |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 9 | 9 | 9 | 9 | 9 | 9 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 18 | ··· | 18 | 18 | ··· | 18 | 36 | ··· | 36 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 54 | 54 | 2 | ··· | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 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 | S3 | S3 | Q8 | D6 | D6 | D6 | D6 | D9 | Dic6 | Dic6 | D18 | D18 | Dic18 | S32 | C32⋊2Q8 | C2×S32 | S3×D9 | C9⋊Dic6 | C2×S3×D9 |
| kernel | C2×C9⋊Dic6 | C9⋊Dic6 | C6×Dic9 | Dic3×C18 | C2×C9⋊Dic3 | C2×Dic9 | C6×Dic3 | C3×C18 | Dic9 | C2×C18 | C3×Dic3 | C62 | C2×Dic3 | C18 | C3×C6 | Dic3 | C2×C6 | C6 | C2×C6 | C6 | C6 | C22 | C2 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 3 | 4 | 4 | 6 | 3 | 12 | 1 | 2 | 1 | 3 | 6 | 3 |
Matrix representation of C2×C9⋊Dic6 ►in GL6(𝔽37)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 36 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 0 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 27 |
| 0 | 0 | 0 | 0 | 3 | 11 |
| 20 | 2 | 0 | 0 | 0 | 0 |
| 3 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 | 0 | 0 |
| 0 | 0 | 1 | 36 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 0 |
| 0 | 0 | 0 | 0 | 35 | 1 |
| 29 | 15 | 0 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,3,0,0,0,0,27,11],[20,3,0,0,0,0,2,17,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,36,35,0,0,0,0,0,1],[29,8,0,0,0,0,15,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C2×C9⋊Dic6 in GAP, Magma, Sage, TeX
C_2\times C_9\rtimes {\rm Dic}_6 % in TeX
G:=Group("C2xC9:Dic6"); // GroupNames label
G:=SmallGroup(432,303);
// by ID
G=gap.SmallGroup(432,303);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,64,3091,662,4037,7069]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^9=c^12=1,d^2=c^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations