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## G = C3×Q8.C18order 432 = 24·33

### Direct product of C3 and Q8.C18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C3×Q8.C18
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8×C32 — C3×Q8⋊C9 — C3×Q8.C18
 Lower central Q8 — C3×Q8.C18
 Upper central C1 — C3×C12

Generators and relations for C3×Q8.C18
G = < a,b,c,d | a3=b4=1, c2=d18=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=bc, dcd-1=b >

Subgroups: 194 in 80 conjugacy classes, 38 normal (17 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, D4, Q8, C9, C32, C12, C12, C12, C2×C6, C4○D4, C18, C3×C6, C3×C6, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C3×C12, C3×C12, C62, C3×C4○D4, C3×C4○D4, C3×C18, Q8⋊C9, C6×C12, D4×C32, Q8×C32, C3×C36, Q8.C18, C32×C4○D4, C3×Q8⋊C9, C3×Q8.C18
Quotients: C1, C2, C3, C6, C9, C32, A4, C18, C3×C6, C2×A4, C3×C9, C3.A4, C3×A4, C4.A4, C3×C18, C2×C3.A4, C6×A4, C3×C3.A4, Q8.C18, C3×C4.A4, C6×C3.A4, C3×Q8.C18

Smallest permutation representation of C3×Q8.C18
On 216 points
Generators in S216
(1 197 150)(2 198 151)(3 199 152)(4 200 153)(5 201 154)(6 202 155)(7 203 156)(8 204 157)(9 205 158)(10 206 159)(11 207 160)(12 208 161)(13 209 162)(14 210 163)(15 211 164)(16 212 165)(17 213 166)(18 214 167)(19 215 168)(20 216 169)(21 181 170)(22 182 171)(23 183 172)(24 184 173)(25 185 174)(26 186 175)(27 187 176)(28 188 177)(29 189 178)(30 190 179)(31 191 180)(32 192 145)(33 193 146)(34 194 147)(35 195 148)(36 196 149)(37 80 135)(38 81 136)(39 82 137)(40 83 138)(41 84 139)(42 85 140)(43 86 141)(44 87 142)(45 88 143)(46 89 144)(47 90 109)(48 91 110)(49 92 111)(50 93 112)(51 94 113)(52 95 114)(53 96 115)(54 97 116)(55 98 117)(56 99 118)(57 100 119)(58 101 120)(59 102 121)(60 103 122)(61 104 123)(62 105 124)(63 106 125)(64 107 126)(65 108 127)(66 73 128)(67 74 129)(68 75 130)(69 76 131)(70 77 132)(71 78 133)(72 79 134)
(1 135 19 117)(2 11 20 29)(3 110 21 128)(4 138 22 120)(5 14 23 32)(6 113 24 131)(7 141 25 123)(8 17 26 35)(9 116 27 134)(10 144 28 126)(12 119 30 137)(13 111 31 129)(15 122 33 140)(16 114 34 132)(18 125 36 143)(37 215 55 197)(38 65 56 47)(39 208 57 190)(40 182 58 200)(41 68 59 50)(42 211 60 193)(43 185 61 203)(44 71 62 53)(45 214 63 196)(46 188 64 206)(48 181 66 199)(49 191 67 209)(51 184 69 202)(52 194 70 212)(54 187 72 205)(73 152 91 170)(74 162 92 180)(75 102 93 84)(76 155 94 173)(77 165 95 147)(78 105 96 87)(79 158 97 176)(80 168 98 150)(81 108 99 90)(82 161 100 179)(83 171 101 153)(85 164 103 146)(86 174 104 156)(88 167 106 149)(89 177 107 159)(109 136 127 118)(112 139 130 121)(115 142 133 124)(145 154 163 172)(148 157 166 175)(151 160 169 178)(183 192 201 210)(186 195 204 213)(189 198 207 216)
(1 144 19 126)(2 136 20 118)(3 12 21 30)(4 111 22 129)(5 139 23 121)(6 15 24 33)(7 114 25 132)(8 142 26 124)(9 18 27 36)(10 117 28 135)(11 109 29 127)(13 120 31 138)(14 112 32 130)(16 123 34 141)(17 115 35 133)(37 206 55 188)(38 216 56 198)(39 66 57 48)(40 209 58 191)(41 183 59 201)(42 69 60 51)(43 212 61 194)(44 186 62 204)(45 72 63 54)(46 215 64 197)(47 189 65 207)(49 182 67 200)(50 192 68 210)(52 185 70 203)(53 195 71 213)(73 100 91 82)(74 153 92 171)(75 163 93 145)(76 103 94 85)(77 156 95 174)(78 166 96 148)(79 106 97 88)(80 159 98 177)(81 169 99 151)(83 162 101 180)(84 172 102 154)(86 165 104 147)(87 175 105 157)(89 168 107 150)(90 178 108 160)(110 137 128 119)(113 140 131 122)(116 143 134 125)(146 155 164 173)(149 158 167 176)(152 161 170 179)(181 190 199 208)(184 193 202 211)(187 196 205 214)
(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)

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

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

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

126 conjugacy classes

 class 1 2A 2B 3A ··· 3H 4A 4B 4C 6A ··· 6H 6I ··· 6P 9A ··· 9R 12A ··· 12P 12Q ··· 12X 18A ··· 18R 36A ··· 36AJ order 1 2 2 3 ··· 3 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 6 1 ··· 1 1 1 6 1 ··· 1 6 ··· 6 4 ··· 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4

126 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 type + + + + image C1 C2 C3 C3 C6 C6 C9 C18 C4.A4 Q8.C18 C3×C4.A4 A4 C2×A4 C3.A4 C3×A4 C2×C3.A4 C6×A4 kernel C3×Q8.C18 C3×Q8⋊C9 Q8.C18 C32×C4○D4 Q8⋊C9 Q8×C32 C3×C4○D4 C3×Q8 C32 C3 C3 C3×C12 C3×C6 C12 C12 C6 C6 # reps 1 1 6 2 6 2 18 18 6 36 12 1 1 6 2 6 2

Matrix representation of C3×Q8.C18 in GL3(𝔽37) generated by

 10 0 0 0 1 0 0 0 1
,
 1 0 0 0 1 1 0 35 36
,
 1 0 0 0 6 0 0 25 31
,
 27 0 0 0 28 4 0 29 29
G:=sub<GL(3,GF(37))| [10,0,0,0,1,0,0,0,1],[1,0,0,0,1,35,0,1,36],[1,0,0,0,6,25,0,0,31],[27,0,0,0,28,29,0,4,29] >;

C3×Q8.C18 in GAP, Magma, Sage, TeX

C_3\times Q_8.C_{18}
% in TeX

G:=Group("C3xQ8.C18");
// GroupNames label

G:=SmallGroup(432,337);
// by ID

G=gap.SmallGroup(432,337);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,134,1901,172,3414,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^4=1,c^2=d^18=b^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d^-1=b*c,d*c*d^-1=b>;
// generators/relations

׿
×
𝔽