Copied to
clipboard

## G = C2×C9⋊Dic3order 216 = 23·33

### Direct product of C2 and C9⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — C2×C9⋊Dic3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9⋊Dic3 — C2×C9⋊Dic3
 Lower central C3×C9 — C2×C9⋊Dic3
 Upper central C1 — C22

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, C3, C3, C4, C22, C6, C6, C2×C4, C9, C32, Dic3, C2×C6, C2×C6, C18, C3×C6, C3×C6, C2×Dic3, C3×C9, Dic9, C2×C18, C3⋊Dic3, C62, C3×C18, C3×C18, C2×Dic9, C2×C3⋊Dic3, C9⋊Dic3, C6×C18, C2×C9⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, D9, C3⋊S3, C2×Dic3, Dic9, D18, C3⋊Dic3, C2×C3⋊S3, C9⋊S3, C2×Dic9, C2×C3⋊Dic3, C9⋊Dic3, 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)(10 172)(11 173)(12 174)(13 175)(14 176)(15 177)(16 178)(17 179)(18 180)(19 168)(20 169)(21 170)(22 171)(23 163)(24 164)(25 165)(26 166)(27 167)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 73)(37 88)(38 89)(39 90)(40 82)(41 83)(42 84)(43 85)(44 86)(45 87)(46 100)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 118)(63 119)(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 40 70 29 55)(2 49 41 71 30 56)(3 50 42 72 31 57)(4 51 43 64 32 58)(5 52 44 65 33 59)(6 53 45 66 34 60)(7 54 37 67 35 61)(8 46 38 68 36 62)(9 47 39 69 28 63)(10 189 26 199 216 193)(11 181 27 200 208 194)(12 182 19 201 209 195)(13 183 20 202 210 196)(14 184 21 203 211 197)(15 185 22 204 212 198)(16 186 23 205 213 190)(17 187 24 206 214 191)(18 188 25 207 215 192)(73 118 95 100 89 111)(74 119 96 101 90 112)(75 120 97 102 82 113)(76 121 98 103 83 114)(77 122 99 104 84 115)(78 123 91 105 85 116)(79 124 92 106 86 117)(80 125 93 107 87 109)(81 126 94 108 88 110)(127 167 146 154 140 173)(128 168 147 155 141 174)(129 169 148 156 142 175)(130 170 149 157 143 176)(131 171 150 158 144 177)(132 163 151 159 136 178)(133 164 152 160 137 179)(134 165 153 161 138 180)(135 166 145 162 139 172)
(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 61 199 37)(11 60 200 45)(12 59 201 44)(13 58 202 43)(14 57 203 42)(15 56 204 41)(16 55 205 40)(17 63 206 39)(18 62 207 38)(28 214 47 187)(29 213 48 186)(30 212 49 185)(31 211 50 184)(32 210 51 183)(33 209 52 182)(34 208 53 181)(35 216 54 189)(36 215 46 188)(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)(82 178 120 151)(83 177 121 150)(84 176 122 149)(85 175 123 148)(86 174 124 147)(87 173 125 146)(88 172 126 145)(89 180 118 153)(90 179 119 152)(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)(10,172)(11,173)(12,174)(13,175)(14,176)(15,177)(16,178)(17,179)(18,180)(19,168)(20,169)(21,170)(22,171)(23,163)(24,164)(25,165)(26,166)(27,167)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,73)(37,88)(38,89)(39,90)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,118)(63,119)(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), (1,48,40,70,29,55)(2,49,41,71,30,56)(3,50,42,72,31,57)(4,51,43,64,32,58)(5,52,44,65,33,59)(6,53,45,66,34,60)(7,54,37,67,35,61)(8,46,38,68,36,62)(9,47,39,69,28,63)(10,189,26,199,216,193)(11,181,27,200,208,194)(12,182,19,201,209,195)(13,183,20,202,210,196)(14,184,21,203,211,197)(15,185,22,204,212,198)(16,186,23,205,213,190)(17,187,24,206,214,191)(18,188,25,207,215,192)(73,118,95,100,89,111)(74,119,96,101,90,112)(75,120,97,102,82,113)(76,121,98,103,83,114)(77,122,99,104,84,115)(78,123,91,105,85,116)(79,124,92,106,86,117)(80,125,93,107,87,109)(81,126,94,108,88,110)(127,167,146,154,140,173)(128,168,147,155,141,174)(129,169,148,156,142,175)(130,170,149,157,143,176)(131,171,150,158,144,177)(132,163,151,159,136,178)(133,164,152,160,137,179)(134,165,153,161,138,180)(135,166,145,162,139,172), (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)(10,61,199,37)(11,60,200,45)(12,59,201,44)(13,58,202,43)(14,57,203,42)(15,56,204,41)(16,55,205,40)(17,63,206,39)(18,62,207,38)(28,214,47,187)(29,213,48,186)(30,212,49,185)(31,211,50,184)(32,210,51,183)(33,209,52,182)(34,208,53,181)(35,216,54,189)(36,215,46,188)(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)(82,178,120,151)(83,177,121,150)(84,176,122,149)(85,175,123,148)(86,174,124,147)(87,173,125,146)(88,172,126,145)(89,180,118,153)(90,179,119,152)(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)(10,172)(11,173)(12,174)(13,175)(14,176)(15,177)(16,178)(17,179)(18,180)(19,168)(20,169)(21,170)(22,171)(23,163)(24,164)(25,165)(26,166)(27,167)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,73)(37,88)(38,89)(39,90)(40,82)(41,83)(42,84)(43,85)(44,86)(45,87)(46,100)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,120)(56,121)(57,122)(58,123)(59,124)(60,125)(61,126)(62,118)(63,119)(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), (1,48,40,70,29,55)(2,49,41,71,30,56)(3,50,42,72,31,57)(4,51,43,64,32,58)(5,52,44,65,33,59)(6,53,45,66,34,60)(7,54,37,67,35,61)(8,46,38,68,36,62)(9,47,39,69,28,63)(10,189,26,199,216,193)(11,181,27,200,208,194)(12,182,19,201,209,195)(13,183,20,202,210,196)(14,184,21,203,211,197)(15,185,22,204,212,198)(16,186,23,205,213,190)(17,187,24,206,214,191)(18,188,25,207,215,192)(73,118,95,100,89,111)(74,119,96,101,90,112)(75,120,97,102,82,113)(76,121,98,103,83,114)(77,122,99,104,84,115)(78,123,91,105,85,116)(79,124,92,106,86,117)(80,125,93,107,87,109)(81,126,94,108,88,110)(127,167,146,154,140,173)(128,168,147,155,141,174)(129,169,148,156,142,175)(130,170,149,157,143,176)(131,171,150,158,144,177)(132,163,151,159,136,178)(133,164,152,160,137,179)(134,165,153,161,138,180)(135,166,145,162,139,172), (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)(10,61,199,37)(11,60,200,45)(12,59,201,44)(13,58,202,43)(14,57,203,42)(15,56,204,41)(16,55,205,40)(17,63,206,39)(18,62,207,38)(28,214,47,187)(29,213,48,186)(30,212,49,185)(31,211,50,184)(32,210,51,183)(33,209,52,182)(34,208,53,181)(35,216,54,189)(36,215,46,188)(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)(82,178,120,151)(83,177,121,150)(84,176,122,149)(85,175,123,148)(86,174,124,147)(87,173,125,146)(88,172,126,145)(89,180,118,153)(90,179,119,152)(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),(10,172),(11,173),(12,174),(13,175),(14,176),(15,177),(16,178),(17,179),(18,180),(19,168),(20,169),(21,170),(22,171),(23,163),(24,164),(25,165),(26,166),(27,167),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,73),(37,88),(38,89),(39,90),(40,82),(41,83),(42,84),(43,85),(44,86),(45,87),(46,100),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,120),(56,121),(57,122),(58,123),(59,124),(60,125),(61,126),(62,118),(63,119),(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)], [(1,48,40,70,29,55),(2,49,41,71,30,56),(3,50,42,72,31,57),(4,51,43,64,32,58),(5,52,44,65,33,59),(6,53,45,66,34,60),(7,54,37,67,35,61),(8,46,38,68,36,62),(9,47,39,69,28,63),(10,189,26,199,216,193),(11,181,27,200,208,194),(12,182,19,201,209,195),(13,183,20,202,210,196),(14,184,21,203,211,197),(15,185,22,204,212,198),(16,186,23,205,213,190),(17,187,24,206,214,191),(18,188,25,207,215,192),(73,118,95,100,89,111),(74,119,96,101,90,112),(75,120,97,102,82,113),(76,121,98,103,83,114),(77,122,99,104,84,115),(78,123,91,105,85,116),(79,124,92,106,86,117),(80,125,93,107,87,109),(81,126,94,108,88,110),(127,167,146,154,140,173),(128,168,147,155,141,174),(129,169,148,156,142,175),(130,170,149,157,143,176),(131,171,150,158,144,177),(132,163,151,159,136,178),(133,164,152,160,137,179),(134,165,153,161,138,180),(135,166,145,162,139,172)], [(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),(10,61,199,37),(11,60,200,45),(12,59,201,44),(13,58,202,43),(14,57,203,42),(15,56,204,41),(16,55,205,40),(17,63,206,39),(18,62,207,38),(28,214,47,187),(29,213,48,186),(30,212,49,185),(31,211,50,184),(32,210,51,183),(33,209,52,182),(34,208,53,181),(35,216,54,189),(36,215,46,188),(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),(82,178,120,151),(83,177,121,150),(84,176,122,149),(85,175,123,148),(86,174,124,147),(87,173,125,146),(88,172,126,145),(89,180,118,153),(90,179,119,152),(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 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6L 9A ··· 9I 18A ··· 18AA order 1 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 9 ··· 9 18 ··· 18 size 1 1 1 1 2 2 2 2 27 27 27 27 2 ··· 2 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + - + + - + image C1 C2 C2 C4 S3 S3 Dic3 D6 Dic3 D6 D9 Dic9 D18 kernel C2×C9⋊Dic3 C9⋊Dic3 C6×C18 C3×C18 C2×C18 C62 C18 C18 C3×C6 C3×C6 C2×C6 C6 C6 # reps 1 2 1 4 3 1 6 3 2 1 9 18 9

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

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 10 0 0 0 10 26 0 0 0 0 11 20 0 0 17 31
,
 27 0 0 0 27 11 0 0 0 0 36 0 0 0 0 36
,
 14 15 0 0 14 23 0 0 0 0 9 29 0 0 1 28
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