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## G = C2×C6.D18order 432 = 24·33

### Direct product of C2 and C6.D18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C2×C6.D18
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C2×C9⋊S3 — C22×C9⋊S3 — C2×C6.D18
 Lower central C3×C9 — C3×C18 — C2×C6.D18
 Upper central C1 — C22 — C23

Generators and relations for C2×C6.D18
G = < a,b,c,d | a2=b6=c18=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 1604 in 270 conjugacy classes, 91 normal (19 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C23, C9, C32, Dic3, D6, C2×C6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C3⋊D4, C22×S3, C22×C6, C22×C6, C3×C9, Dic9, D18, C2×C18, C2×C18, C3⋊Dic3, C2×C3⋊S3, C62, C62, C62, C2×C3⋊D4, C9⋊S3, C3×C18, C3×C18, C3×C18, C2×Dic9, C9⋊D4, C22×D9, C22×C18, C2×C3⋊Dic3, C327D4, C22×C3⋊S3, C2×C62, C9⋊Dic3, C2×C9⋊S3, C2×C9⋊S3, C6×C18, C6×C18, C6×C18, C2×C9⋊D4, C2×C327D4, C2×C9⋊Dic3, C6.D18, C22×C9⋊S3, C2×C6×C18, C2×C6.D18
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, C3⋊S3, C3⋊D4, C22×S3, D18, C2×C3⋊S3, C2×C3⋊D4, C9⋊S3, C9⋊D4, C22×D9, C327D4, C22×C3⋊S3, C2×C9⋊S3, C2×C9⋊D4, C2×C327D4, C6.D18, C22×C9⋊S3, C2×C6.D18

Smallest permutation representation of C2×C6.D18
On 216 points
Generators in S216
(1 10)(2 11)(3 12)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 73)(33 74)(34 75)(35 76)(36 77)(37 124)(38 125)(39 126)(40 109)(41 110)(42 111)(43 112)(44 113)(45 114)(46 115)(47 116)(48 117)(49 118)(50 119)(51 120)(52 121)(53 122)(54 123)(55 150)(56 151)(57 152)(58 153)(59 154)(60 155)(61 156)(62 157)(63 158)(64 159)(65 160)(66 161)(67 162)(68 145)(69 146)(70 147)(71 148)(72 149)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(127 136)(128 137)(129 138)(130 139)(131 140)(132 141)(133 142)(134 143)(135 144)(163 172)(164 173)(165 174)(166 175)(167 176)(168 177)(169 178)(170 179)(171 180)(181 190)(182 191)(183 192)(184 193)(185 194)(186 195)(187 196)(188 197)(189 198)(199 208)(200 209)(201 210)(202 211)(203 212)(204 213)(205 214)(206 215)(207 216)
(1 142 163 187 212 93)(2 143 164 188 213 94)(3 144 165 189 214 95)(4 127 166 190 215 96)(5 128 167 191 216 97)(6 129 168 192 199 98)(7 130 169 193 200 99)(8 131 170 194 201 100)(9 132 171 195 202 101)(10 133 172 196 203 102)(11 134 173 197 204 103)(12 135 174 198 205 104)(13 136 175 181 206 105)(14 137 176 182 207 106)(15 138 177 183 208 107)(16 139 178 184 209 108)(17 140 179 185 210 91)(18 141 180 186 211 92)(19 124 152 87 46 66)(20 125 153 88 47 67)(21 126 154 89 48 68)(22 109 155 90 49 69)(23 110 156 73 50 70)(24 111 157 74 51 71)(25 112 158 75 52 72)(26 113 159 76 53 55)(27 114 160 77 54 56)(28 115 161 78 37 57)(29 116 162 79 38 58)(30 117 145 80 39 59)(31 118 146 81 40 60)(32 119 147 82 41 61)(33 120 148 83 42 62)(34 121 149 84 43 63)(35 122 150 85 44 64)(36 123 151 86 45 65)
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162)(163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)
(1 154 187 68)(2 67 188 153)(3 152 189 66)(4 65 190 151)(5 150 191 64)(6 63 192 149)(7 148 193 62)(8 61 194 147)(9 146 195 60)(10 59 196 145)(11 162 197 58)(12 57 198 161)(13 160 181 56)(14 55 182 159)(15 158 183 72)(16 71 184 157)(17 156 185 70)(18 69 186 155)(19 95 87 165)(20 164 88 94)(21 93 89 163)(22 180 90 92)(23 91 73 179)(24 178 74 108)(25 107 75 177)(26 176 76 106)(27 105 77 175)(28 174 78 104)(29 103 79 173)(30 172 80 102)(31 101 81 171)(32 170 82 100)(33 99 83 169)(34 168 84 98)(35 97 85 167)(36 166 86 96)(37 205 115 135)(38 134 116 204)(39 203 117 133)(40 132 118 202)(41 201 119 131)(42 130 120 200)(43 199 121 129)(44 128 122 216)(45 215 123 127)(46 144 124 214)(47 213 125 143)(48 142 126 212)(49 211 109 141)(50 140 110 210)(51 209 111 139)(52 138 112 208)(53 207 113 137)(54 136 114 206)

G:=sub<Sym(216)| (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,73)(33,74)(34,75)(35,76)(36,77)(37,124)(38,125)(39,126)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,157)(63,158)(64,159)(65,160)(66,161)(67,162)(68,145)(69,146)(70,147)(71,148)(72,149)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141)(133,142)(134,143)(135,144)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(181,190)(182,191)(183,192)(184,193)(185,194)(186,195)(187,196)(188,197)(189,198)(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216), (1,142,163,187,212,93)(2,143,164,188,213,94)(3,144,165,189,214,95)(4,127,166,190,215,96)(5,128,167,191,216,97)(6,129,168,192,199,98)(7,130,169,193,200,99)(8,131,170,194,201,100)(9,132,171,195,202,101)(10,133,172,196,203,102)(11,134,173,197,204,103)(12,135,174,198,205,104)(13,136,175,181,206,105)(14,137,176,182,207,106)(15,138,177,183,208,107)(16,139,178,184,209,108)(17,140,179,185,210,91)(18,141,180,186,211,92)(19,124,152,87,46,66)(20,125,153,88,47,67)(21,126,154,89,48,68)(22,109,155,90,49,69)(23,110,156,73,50,70)(24,111,157,74,51,71)(25,112,158,75,52,72)(26,113,159,76,53,55)(27,114,160,77,54,56)(28,115,161,78,37,57)(29,116,162,79,38,58)(30,117,145,80,39,59)(31,118,146,81,40,60)(32,119,147,82,41,61)(33,120,148,83,42,62)(34,121,149,84,43,63)(35,122,150,85,44,64)(36,123,151,86,45,65), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,154,187,68)(2,67,188,153)(3,152,189,66)(4,65,190,151)(5,150,191,64)(6,63,192,149)(7,148,193,62)(8,61,194,147)(9,146,195,60)(10,59,196,145)(11,162,197,58)(12,57,198,161)(13,160,181,56)(14,55,182,159)(15,158,183,72)(16,71,184,157)(17,156,185,70)(18,69,186,155)(19,95,87,165)(20,164,88,94)(21,93,89,163)(22,180,90,92)(23,91,73,179)(24,178,74,108)(25,107,75,177)(26,176,76,106)(27,105,77,175)(28,174,78,104)(29,103,79,173)(30,172,80,102)(31,101,81,171)(32,170,82,100)(33,99,83,169)(34,168,84,98)(35,97,85,167)(36,166,86,96)(37,205,115,135)(38,134,116,204)(39,203,117,133)(40,132,118,202)(41,201,119,131)(42,130,120,200)(43,199,121,129)(44,128,122,216)(45,215,123,127)(46,144,124,214)(47,213,125,143)(48,142,126,212)(49,211,109,141)(50,140,110,210)(51,209,111,139)(52,138,112,208)(53,207,113,137)(54,136,114,206)>;

G:=Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,84)(26,85)(27,86)(28,87)(29,88)(30,89)(31,90)(32,73)(33,74)(34,75)(35,76)(36,77)(37,124)(38,125)(39,126)(40,109)(41,110)(42,111)(43,112)(44,113)(45,114)(46,115)(47,116)(48,117)(49,118)(50,119)(51,120)(52,121)(53,122)(54,123)(55,150)(56,151)(57,152)(58,153)(59,154)(60,155)(61,156)(62,157)(63,158)(64,159)(65,160)(66,161)(67,162)(68,145)(69,146)(70,147)(71,148)(72,149)(91,100)(92,101)(93,102)(94,103)(95,104)(96,105)(97,106)(98,107)(99,108)(127,136)(128,137)(129,138)(130,139)(131,140)(132,141)(133,142)(134,143)(135,144)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(181,190)(182,191)(183,192)(184,193)(185,194)(186,195)(187,196)(188,197)(189,198)(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216), (1,142,163,187,212,93)(2,143,164,188,213,94)(3,144,165,189,214,95)(4,127,166,190,215,96)(5,128,167,191,216,97)(6,129,168,192,199,98)(7,130,169,193,200,99)(8,131,170,194,201,100)(9,132,171,195,202,101)(10,133,172,196,203,102)(11,134,173,197,204,103)(12,135,174,198,205,104)(13,136,175,181,206,105)(14,137,176,182,207,106)(15,138,177,183,208,107)(16,139,178,184,209,108)(17,140,179,185,210,91)(18,141,180,186,211,92)(19,124,152,87,46,66)(20,125,153,88,47,67)(21,126,154,89,48,68)(22,109,155,90,49,69)(23,110,156,73,50,70)(24,111,157,74,51,71)(25,112,158,75,52,72)(26,113,159,76,53,55)(27,114,160,77,54,56)(28,115,161,78,37,57)(29,116,162,79,38,58)(30,117,145,80,39,59)(31,118,146,81,40,60)(32,119,147,82,41,61)(33,120,148,83,42,62)(34,121,149,84,43,63)(35,122,150,85,44,64)(36,123,151,86,45,65), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162)(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216), (1,154,187,68)(2,67,188,153)(3,152,189,66)(4,65,190,151)(5,150,191,64)(6,63,192,149)(7,148,193,62)(8,61,194,147)(9,146,195,60)(10,59,196,145)(11,162,197,58)(12,57,198,161)(13,160,181,56)(14,55,182,159)(15,158,183,72)(16,71,184,157)(17,156,185,70)(18,69,186,155)(19,95,87,165)(20,164,88,94)(21,93,89,163)(22,180,90,92)(23,91,73,179)(24,178,74,108)(25,107,75,177)(26,176,76,106)(27,105,77,175)(28,174,78,104)(29,103,79,173)(30,172,80,102)(31,101,81,171)(32,170,82,100)(33,99,83,169)(34,168,84,98)(35,97,85,167)(36,166,86,96)(37,205,115,135)(38,134,116,204)(39,203,117,133)(40,132,118,202)(41,201,119,131)(42,130,120,200)(43,199,121,129)(44,128,122,216)(45,215,123,127)(46,144,124,214)(47,213,125,143)(48,142,126,212)(49,211,109,141)(50,140,110,210)(51,209,111,139)(52,138,112,208)(53,207,113,137)(54,136,114,206) );

G=PermutationGroup([[(1,10),(2,11),(3,12),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,84),(26,85),(27,86),(28,87),(29,88),(30,89),(31,90),(32,73),(33,74),(34,75),(35,76),(36,77),(37,124),(38,125),(39,126),(40,109),(41,110),(42,111),(43,112),(44,113),(45,114),(46,115),(47,116),(48,117),(49,118),(50,119),(51,120),(52,121),(53,122),(54,123),(55,150),(56,151),(57,152),(58,153),(59,154),(60,155),(61,156),(62,157),(63,158),(64,159),(65,160),(66,161),(67,162),(68,145),(69,146),(70,147),(71,148),(72,149),(91,100),(92,101),(93,102),(94,103),(95,104),(96,105),(97,106),(98,107),(99,108),(127,136),(128,137),(129,138),(130,139),(131,140),(132,141),(133,142),(134,143),(135,144),(163,172),(164,173),(165,174),(166,175),(167,176),(168,177),(169,178),(170,179),(171,180),(181,190),(182,191),(183,192),(184,193),(185,194),(186,195),(187,196),(188,197),(189,198),(199,208),(200,209),(201,210),(202,211),(203,212),(204,213),(205,214),(206,215),(207,216)], [(1,142,163,187,212,93),(2,143,164,188,213,94),(3,144,165,189,214,95),(4,127,166,190,215,96),(5,128,167,191,216,97),(6,129,168,192,199,98),(7,130,169,193,200,99),(8,131,170,194,201,100),(9,132,171,195,202,101),(10,133,172,196,203,102),(11,134,173,197,204,103),(12,135,174,198,205,104),(13,136,175,181,206,105),(14,137,176,182,207,106),(15,138,177,183,208,107),(16,139,178,184,209,108),(17,140,179,185,210,91),(18,141,180,186,211,92),(19,124,152,87,46,66),(20,125,153,88,47,67),(21,126,154,89,48,68),(22,109,155,90,49,69),(23,110,156,73,50,70),(24,111,157,74,51,71),(25,112,158,75,52,72),(26,113,159,76,53,55),(27,114,160,77,54,56),(28,115,161,78,37,57),(29,116,162,79,38,58),(30,117,145,80,39,59),(31,118,146,81,40,60),(32,119,147,82,41,61),(33,120,148,83,42,62),(34,121,149,84,43,63),(35,122,150,85,44,64),(36,123,151,86,45,65)], [(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),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162),(163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)], [(1,154,187,68),(2,67,188,153),(3,152,189,66),(4,65,190,151),(5,150,191,64),(6,63,192,149),(7,148,193,62),(8,61,194,147),(9,146,195,60),(10,59,196,145),(11,162,197,58),(12,57,198,161),(13,160,181,56),(14,55,182,159),(15,158,183,72),(16,71,184,157),(17,156,185,70),(18,69,186,155),(19,95,87,165),(20,164,88,94),(21,93,89,163),(22,180,90,92),(23,91,73,179),(24,178,74,108),(25,107,75,177),(26,176,76,106),(27,105,77,175),(28,174,78,104),(29,103,79,173),(30,172,80,102),(31,101,81,171),(32,170,82,100),(33,99,83,169),(34,168,84,98),(35,97,85,167),(36,166,86,96),(37,205,115,135),(38,134,116,204),(39,203,117,133),(40,132,118,202),(41,201,119,131),(42,130,120,200),(43,199,121,129),(44,128,122,216),(45,215,123,127),(46,144,124,214),(47,213,125,143),(48,142,126,212),(49,211,109,141),(50,140,110,210),(51,209,111,139),(52,138,112,208),(53,207,113,137),(54,136,114,206)]])

114 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A ··· 6AB 9A ··· 9I 18A ··· 18BK order 1 2 2 2 2 2 2 2 3 3 3 3 4 4 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 2 2 54 54 2 2 2 2 54 54 2 ··· 2 2 ··· 2 2 ··· 2

114 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 S3 D4 D6 D6 D9 C3⋊D4 C3⋊D4 D18 C9⋊D4 kernel C2×C6.D18 C2×C9⋊Dic3 C6.D18 C22×C9⋊S3 C2×C6×C18 C22×C18 C2×C62 C3×C18 C2×C18 C62 C22×C6 C18 C3×C6 C2×C6 C6 # reps 1 1 4 1 1 3 1 2 9 3 9 12 4 27 36

Matrix representation of C2×C6.D18 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 36 0 0 0 0 36
,
 27 0 0 0 0 11 0 0 0 0 36 0 0 0 0 36
,
 12 0 0 0 0 3 0 0 0 0 30 0 0 0 0 16
,
 0 34 0 0 25 0 0 0 0 0 0 21 0 0 7 0
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,36,0,0,0,0,36],[27,0,0,0,0,11,0,0,0,0,36,0,0,0,0,36],[12,0,0,0,0,3,0,0,0,0,30,0,0,0,0,16],[0,25,0,0,34,0,0,0,0,0,0,7,0,0,21,0] >;

C2×C6.D18 in GAP, Magma, Sage, TeX

C_2\times C_6.D_{18}
% in TeX

G:=Group("C2xC6.D18");
// GroupNames label

G:=SmallGroup(432,397);
// by ID

G=gap.SmallGroup(432,397);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,6164,662,4037,14118]);
// Polycyclic

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

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