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G = C2×C9⋊Dic3order 216 = 23·33

Direct product of C2 and C9⋊Dic3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2×C9⋊Dic3, C6⋊Dic9, C18⋊Dic3, C6.17D18, C18.17D6, C62.11S3, (C3×C18)⋊3C4, (C2×C6).5D9, (C2×C18).5S3, (C6×C18).3C2, C22.(C9⋊S3), C92(C2×Dic3), C32(C2×Dic9), (C3×C6).53D6, (C3×C6).9Dic3, C6.2(C3⋊Dic3), (C3×C18).21C22, C32.4(C2×Dic3), (C3×C9)⋊9(C2×C4), C2.2(C2×C9⋊S3), C3.(C2×C3⋊Dic3), C6.11(C2×C3⋊S3), (C2×C6).5(C3⋊S3), SmallGroup(216,69)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C2×C9⋊Dic3
C1C3C32C3×C9C3×C18C9⋊Dic3 — C2×C9⋊Dic3
C3×C9 — C2×C9⋊Dic3
C1C22

Generators and relations for C2×C9⋊Dic3
 G = < a,b,c,d | a2=b9=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 278 in 80 conjugacy classes, 53 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4 [×2], C22, C6, C6 [×11], C2×C4, C9 [×3], C32, Dic3 [×8], C2×C6, C2×C6 [×3], C18 [×9], C3×C6, C3×C6 [×2], C2×Dic3 [×4], C3×C9, Dic9 [×6], C2×C18 [×3], C3⋊Dic3 [×2], C62, C3×C18, C3×C18 [×2], C2×Dic9 [×3], C2×C3⋊Dic3, C9⋊Dic3 [×2], C6×C18, C2×C9⋊Dic3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, Dic3 [×8], D6 [×4], D9 [×3], C3⋊S3, C2×Dic3 [×4], Dic9 [×6], D18 [×3], C3⋊Dic3 [×2], C2×C3⋊S3, C9⋊S3, C2×Dic9 [×3], C2×C3⋊Dic3, C9⋊Dic3 [×2], C2×C9⋊S3, C2×C9⋊Dic3

Smallest permutation representation of C2×C9⋊Dic3
Regular action on 216 points
Generators in S216
(1 97)(2 98)(3 99)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 179)(11 180)(12 172)(13 173)(14 174)(15 175)(16 176)(17 177)(18 178)(19 168)(20 169)(21 170)(22 171)(23 163)(24 164)(25 165)(26 166)(27 167)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 81)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 121)(56 122)(57 123)(58 124)(59 125)(60 126)(61 118)(62 119)(63 120)(64 116)(65 117)(66 109)(67 110)(68 111)(69 112)(70 113)(71 114)(72 115)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 193)(140 194)(141 195)(142 196)(143 197)(144 198)(145 199)(146 200)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(153 207)(154 208)(155 209)(156 210)(157 211)(158 212)(159 213)(160 214)(161 215)(162 216)
(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 48 35 70 40 62)(2 49 36 71 41 63)(3 50 28 72 42 55)(4 51 29 64 43 56)(5 52 30 65 44 57)(6 53 31 66 45 58)(7 54 32 67 37 59)(8 46 33 68 38 60)(9 47 34 69 39 61)(10 188 25 206 215 192)(11 189 26 207 216 193)(12 181 27 199 208 194)(13 182 19 200 209 195)(14 183 20 201 210 196)(15 184 21 202 211 197)(16 185 22 203 212 198)(17 186 23 204 213 190)(18 187 24 205 214 191)(73 126 95 100 87 111)(74 118 96 101 88 112)(75 119 97 102 89 113)(76 120 98 103 90 114)(77 121 99 104 82 115)(78 122 91 105 83 116)(79 123 92 106 84 117)(80 124 93 107 85 109)(81 125 94 108 86 110)(127 167 145 154 140 172)(128 168 146 155 141 173)(129 169 147 156 142 174)(130 170 148 157 143 175)(131 171 149 158 144 176)(132 163 150 159 136 177)(133 164 151 160 137 178)(134 165 152 161 138 179)(135 166 153 162 139 180)
(1 23 70 190)(2 22 71 198)(3 21 72 197)(4 20 64 196)(5 19 65 195)(6 27 66 194)(7 26 67 193)(8 25 68 192)(9 24 69 191)(10 60 206 33)(11 59 207 32)(12 58 199 31)(13 57 200 30)(14 56 201 29)(15 55 202 28)(16 63 203 36)(17 62 204 35)(18 61 205 34)(37 216 54 189)(38 215 46 188)(39 214 47 187)(40 213 48 186)(41 212 49 185)(42 211 50 184)(43 210 51 183)(44 209 52 182)(45 208 53 181)(73 161 100 134)(74 160 101 133)(75 159 102 132)(76 158 103 131)(77 157 104 130)(78 156 105 129)(79 155 106 128)(80 154 107 127)(81 162 108 135)(82 175 121 148)(83 174 122 147)(84 173 123 146)(85 172 124 145)(86 180 125 153)(87 179 126 152)(88 178 118 151)(89 177 119 150)(90 176 120 149)(91 169 116 142)(92 168 117 141)(93 167 109 140)(94 166 110 139)(95 165 111 138)(96 164 112 137)(97 163 113 136)(98 171 114 144)(99 170 115 143)

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

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

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

C2×C9⋊Dic3 is a maximal subgroup of
Dic3×Dic9  Dic9⋊Dic3  C18.Dic6  Dic3⋊Dic9  D18⋊Dic3  D6⋊Dic9  C6.Dic18  C36⋊Dic3  C6.11D36  C62.127D6  C2×Dic3×D9  C2×S3×Dic9  D18.4D6  C2×C4×C9⋊S3  C36.27D6
C2×C9⋊Dic3 is a maximal quotient of
C36.69D6  C36⋊Dic3  C62.127D6

60 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A···6L9A···9I18A···18AA
order1222333344446···69···918···18
size11112222272727272···22···22···2

60 irreducible representations

dim1111222222222
type+++++-+-++-+
imageC1C2C2C4S3S3Dic3D6Dic3D6D9Dic9D18
kernelC2×C9⋊Dic3C9⋊Dic3C6×C18C3×C18C2×C18C62C18C18C3×C6C3×C6C2×C6C6C6
# reps12143163219189

Matrix representation of C2×C9⋊Dic3 in GL4(𝔽37) generated by

36000
03600
0010
0001
,
10000
102600
001120
001731
,
27000
271100
00360
00036
,
141500
142300
00929
00128
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[10,10,0,0,0,26,0,0,0,0,11,17,0,0,20,31],[27,27,0,0,0,11,0,0,0,0,36,0,0,0,0,36],[14,14,0,0,15,23,0,0,0,0,9,1,0,0,29,28] >;

C2×C9⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xC9:Dic3");
// GroupNames label

G:=SmallGroup(216,69);
// by ID

G=gap.SmallGroup(216,69);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,2115,453,1444,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^9=c^6=1,d^2=c^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=c^-1>;
// generators/relations

׿
×
𝔽