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G = C7×C8.18D4order 448 = 26·7

Direct product of C7 and C8.18D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C4 — C7×C8.18D4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C28 — Q8×C14 — C14×Q16 — C7×C8.18D4
 Lower central C1 — C2 — C2×C4 — C7×C8.18D4
 Upper central C1 — C2×C14 — C22×C28 — C7×C8.18D4

Generators and relations for C7×C8.18D4
G = < a,b,c,d | a7=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 186 in 114 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C8, C2×C4, C2×C4, Q8, C23, C14, C14, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C28, C28, C2×C14, C2×C14, C2×C14, Q8⋊C4, C2.D8, C22⋊Q8, C22×C8, C2×Q16, C56, C56, C2×C28, C2×C28, C7×Q8, C22×C14, C8.18D4, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C2×C56, C2×C56, C7×Q16, C22×C28, Q8×C14, C7×Q8⋊C4, C7×C2.D8, C7×C22⋊Q8, C22×C56, C14×Q16, C7×C8.18D4
Quotients: C1, C2, C22, C7, D4, C23, C14, Q16, C2×D4, C4○D4, C2×C14, C4⋊D4, C2×Q16, C4○D8, C7×D4, C22×C14, C8.18D4, C7×Q16, D4×C14, C7×C4○D4, C7×C4⋊D4, C14×Q16, C7×C4○D8, C7×C8.18D4

Smallest permutation representation of C7×C8.18D4
On 224 points
Generators in S224
(1 117 107 49 99 41 91)(2 118 108 50 100 42 92)(3 119 109 51 101 43 93)(4 120 110 52 102 44 94)(5 113 111 53 103 45 95)(6 114 112 54 104 46 96)(7 115 105 55 97 47 89)(8 116 106 56 98 48 90)(9 33 85 25 77 17 69)(10 34 86 26 78 18 70)(11 35 87 27 79 19 71)(12 36 88 28 80 20 72)(13 37 81 29 73 21 65)(14 38 82 30 74 22 66)(15 39 83 31 75 23 67)(16 40 84 32 76 24 68)(57 218 164 210 156 202 148)(58 219 165 211 157 203 149)(59 220 166 212 158 204 150)(60 221 167 213 159 205 151)(61 222 168 214 160 206 152)(62 223 161 215 153 207 145)(63 224 162 216 154 208 146)(64 217 163 209 155 201 147)(121 175 199 137 191 129 183)(122 176 200 138 192 130 184)(123 169 193 139 185 131 177)(124 170 194 140 186 132 178)(125 171 195 141 187 133 179)(126 172 196 142 188 134 180)(127 173 197 143 189 135 181)(128 174 198 144 190 136 182)
(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)(217 218 219 220 221 222 223 224)
(1 170 11 151)(2 169 12 150)(3 176 13 149)(4 175 14 148)(5 174 15 147)(6 173 16 146)(7 172 9 145)(8 171 10 152)(17 153 47 180)(18 160 48 179)(19 159 41 178)(20 158 42 177)(21 157 43 184)(22 156 44 183)(23 155 45 182)(24 154 46 181)(25 161 55 188)(26 168 56 187)(27 167 49 186)(28 166 50 185)(29 165 51 192)(30 164 52 191)(31 163 53 190)(32 162 54 189)(33 62 115 196)(34 61 116 195)(35 60 117 194)(36 59 118 193)(37 58 119 200)(38 57 120 199)(39 64 113 198)(40 63 114 197)(65 203 93 122)(66 202 94 121)(67 201 95 128)(68 208 96 127)(69 207 89 126)(70 206 90 125)(71 205 91 124)(72 204 92 123)(73 211 101 130)(74 210 102 129)(75 209 103 136)(76 216 104 135)(77 215 97 134)(78 214 98 133)(79 213 99 132)(80 212 100 131)(81 219 109 138)(82 218 110 137)(83 217 111 144)(84 224 112 143)(85 223 105 142)(86 222 106 141)(87 221 107 140)(88 220 108 139)
(1 147 5 151)(2 146 6 150)(3 145 7 149)(4 152 8 148)(9 176 13 172)(10 175 14 171)(11 174 15 170)(12 173 16 169)(17 184 21 180)(18 183 22 179)(19 182 23 178)(20 181 24 177)(25 192 29 188)(26 191 30 187)(27 190 31 186)(28 189 32 185)(33 200 37 196)(34 199 38 195)(35 198 39 194)(36 197 40 193)(41 155 45 159)(42 154 46 158)(43 153 47 157)(44 160 48 156)(49 163 53 167)(50 162 54 166)(51 161 55 165)(52 168 56 164)(57 120 61 116)(58 119 62 115)(59 118 63 114)(60 117 64 113)(65 126 69 122)(66 125 70 121)(67 124 71 128)(68 123 72 127)(73 134 77 130)(74 133 78 129)(75 132 79 136)(76 131 80 135)(81 142 85 138)(82 141 86 137)(83 140 87 144)(84 139 88 143)(89 203 93 207)(90 202 94 206)(91 201 95 205)(92 208 96 204)(97 211 101 215)(98 210 102 214)(99 209 103 213)(100 216 104 212)(105 219 109 223)(106 218 110 222)(107 217 111 221)(108 224 112 220)

G:=sub<Sym(224)| (1,117,107,49,99,41,91)(2,118,108,50,100,42,92)(3,119,109,51,101,43,93)(4,120,110,52,102,44,94)(5,113,111,53,103,45,95)(6,114,112,54,104,46,96)(7,115,105,55,97,47,89)(8,116,106,56,98,48,90)(9,33,85,25,77,17,69)(10,34,86,26,78,18,70)(11,35,87,27,79,19,71)(12,36,88,28,80,20,72)(13,37,81,29,73,21,65)(14,38,82,30,74,22,66)(15,39,83,31,75,23,67)(16,40,84,32,76,24,68)(57,218,164,210,156,202,148)(58,219,165,211,157,203,149)(59,220,166,212,158,204,150)(60,221,167,213,159,205,151)(61,222,168,214,160,206,152)(62,223,161,215,153,207,145)(63,224,162,216,154,208,146)(64,217,163,209,155,201,147)(121,175,199,137,191,129,183)(122,176,200,138,192,130,184)(123,169,193,139,185,131,177)(124,170,194,140,186,132,178)(125,171,195,141,187,133,179)(126,172,196,142,188,134,180)(127,173,197,143,189,135,181)(128,174,198,144,190,136,182), (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)(217,218,219,220,221,222,223,224), (1,170,11,151)(2,169,12,150)(3,176,13,149)(4,175,14,148)(5,174,15,147)(6,173,16,146)(7,172,9,145)(8,171,10,152)(17,153,47,180)(18,160,48,179)(19,159,41,178)(20,158,42,177)(21,157,43,184)(22,156,44,183)(23,155,45,182)(24,154,46,181)(25,161,55,188)(26,168,56,187)(27,167,49,186)(28,166,50,185)(29,165,51,192)(30,164,52,191)(31,163,53,190)(32,162,54,189)(33,62,115,196)(34,61,116,195)(35,60,117,194)(36,59,118,193)(37,58,119,200)(38,57,120,199)(39,64,113,198)(40,63,114,197)(65,203,93,122)(66,202,94,121)(67,201,95,128)(68,208,96,127)(69,207,89,126)(70,206,90,125)(71,205,91,124)(72,204,92,123)(73,211,101,130)(74,210,102,129)(75,209,103,136)(76,216,104,135)(77,215,97,134)(78,214,98,133)(79,213,99,132)(80,212,100,131)(81,219,109,138)(82,218,110,137)(83,217,111,144)(84,224,112,143)(85,223,105,142)(86,222,106,141)(87,221,107,140)(88,220,108,139), (1,147,5,151)(2,146,6,150)(3,145,7,149)(4,152,8,148)(9,176,13,172)(10,175,14,171)(11,174,15,170)(12,173,16,169)(17,184,21,180)(18,183,22,179)(19,182,23,178)(20,181,24,177)(25,192,29,188)(26,191,30,187)(27,190,31,186)(28,189,32,185)(33,200,37,196)(34,199,38,195)(35,198,39,194)(36,197,40,193)(41,155,45,159)(42,154,46,158)(43,153,47,157)(44,160,48,156)(49,163,53,167)(50,162,54,166)(51,161,55,165)(52,168,56,164)(57,120,61,116)(58,119,62,115)(59,118,63,114)(60,117,64,113)(65,126,69,122)(66,125,70,121)(67,124,71,128)(68,123,72,127)(73,134,77,130)(74,133,78,129)(75,132,79,136)(76,131,80,135)(81,142,85,138)(82,141,86,137)(83,140,87,144)(84,139,88,143)(89,203,93,207)(90,202,94,206)(91,201,95,205)(92,208,96,204)(97,211,101,215)(98,210,102,214)(99,209,103,213)(100,216,104,212)(105,219,109,223)(106,218,110,222)(107,217,111,221)(108,224,112,220)>;

G:=Group( (1,117,107,49,99,41,91)(2,118,108,50,100,42,92)(3,119,109,51,101,43,93)(4,120,110,52,102,44,94)(5,113,111,53,103,45,95)(6,114,112,54,104,46,96)(7,115,105,55,97,47,89)(8,116,106,56,98,48,90)(9,33,85,25,77,17,69)(10,34,86,26,78,18,70)(11,35,87,27,79,19,71)(12,36,88,28,80,20,72)(13,37,81,29,73,21,65)(14,38,82,30,74,22,66)(15,39,83,31,75,23,67)(16,40,84,32,76,24,68)(57,218,164,210,156,202,148)(58,219,165,211,157,203,149)(59,220,166,212,158,204,150)(60,221,167,213,159,205,151)(61,222,168,214,160,206,152)(62,223,161,215,153,207,145)(63,224,162,216,154,208,146)(64,217,163,209,155,201,147)(121,175,199,137,191,129,183)(122,176,200,138,192,130,184)(123,169,193,139,185,131,177)(124,170,194,140,186,132,178)(125,171,195,141,187,133,179)(126,172,196,142,188,134,180)(127,173,197,143,189,135,181)(128,174,198,144,190,136,182), (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)(217,218,219,220,221,222,223,224), (1,170,11,151)(2,169,12,150)(3,176,13,149)(4,175,14,148)(5,174,15,147)(6,173,16,146)(7,172,9,145)(8,171,10,152)(17,153,47,180)(18,160,48,179)(19,159,41,178)(20,158,42,177)(21,157,43,184)(22,156,44,183)(23,155,45,182)(24,154,46,181)(25,161,55,188)(26,168,56,187)(27,167,49,186)(28,166,50,185)(29,165,51,192)(30,164,52,191)(31,163,53,190)(32,162,54,189)(33,62,115,196)(34,61,116,195)(35,60,117,194)(36,59,118,193)(37,58,119,200)(38,57,120,199)(39,64,113,198)(40,63,114,197)(65,203,93,122)(66,202,94,121)(67,201,95,128)(68,208,96,127)(69,207,89,126)(70,206,90,125)(71,205,91,124)(72,204,92,123)(73,211,101,130)(74,210,102,129)(75,209,103,136)(76,216,104,135)(77,215,97,134)(78,214,98,133)(79,213,99,132)(80,212,100,131)(81,219,109,138)(82,218,110,137)(83,217,111,144)(84,224,112,143)(85,223,105,142)(86,222,106,141)(87,221,107,140)(88,220,108,139), (1,147,5,151)(2,146,6,150)(3,145,7,149)(4,152,8,148)(9,176,13,172)(10,175,14,171)(11,174,15,170)(12,173,16,169)(17,184,21,180)(18,183,22,179)(19,182,23,178)(20,181,24,177)(25,192,29,188)(26,191,30,187)(27,190,31,186)(28,189,32,185)(33,200,37,196)(34,199,38,195)(35,198,39,194)(36,197,40,193)(41,155,45,159)(42,154,46,158)(43,153,47,157)(44,160,48,156)(49,163,53,167)(50,162,54,166)(51,161,55,165)(52,168,56,164)(57,120,61,116)(58,119,62,115)(59,118,63,114)(60,117,64,113)(65,126,69,122)(66,125,70,121)(67,124,71,128)(68,123,72,127)(73,134,77,130)(74,133,78,129)(75,132,79,136)(76,131,80,135)(81,142,85,138)(82,141,86,137)(83,140,87,144)(84,139,88,143)(89,203,93,207)(90,202,94,206)(91,201,95,205)(92,208,96,204)(97,211,101,215)(98,210,102,214)(99,209,103,213)(100,216,104,212)(105,219,109,223)(106,218,110,222)(107,217,111,221)(108,224,112,220) );

G=PermutationGroup([[(1,117,107,49,99,41,91),(2,118,108,50,100,42,92),(3,119,109,51,101,43,93),(4,120,110,52,102,44,94),(5,113,111,53,103,45,95),(6,114,112,54,104,46,96),(7,115,105,55,97,47,89),(8,116,106,56,98,48,90),(9,33,85,25,77,17,69),(10,34,86,26,78,18,70),(11,35,87,27,79,19,71),(12,36,88,28,80,20,72),(13,37,81,29,73,21,65),(14,38,82,30,74,22,66),(15,39,83,31,75,23,67),(16,40,84,32,76,24,68),(57,218,164,210,156,202,148),(58,219,165,211,157,203,149),(59,220,166,212,158,204,150),(60,221,167,213,159,205,151),(61,222,168,214,160,206,152),(62,223,161,215,153,207,145),(63,224,162,216,154,208,146),(64,217,163,209,155,201,147),(121,175,199,137,191,129,183),(122,176,200,138,192,130,184),(123,169,193,139,185,131,177),(124,170,194,140,186,132,178),(125,171,195,141,187,133,179),(126,172,196,142,188,134,180),(127,173,197,143,189,135,181),(128,174,198,144,190,136,182)], [(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),(217,218,219,220,221,222,223,224)], [(1,170,11,151),(2,169,12,150),(3,176,13,149),(4,175,14,148),(5,174,15,147),(6,173,16,146),(7,172,9,145),(8,171,10,152),(17,153,47,180),(18,160,48,179),(19,159,41,178),(20,158,42,177),(21,157,43,184),(22,156,44,183),(23,155,45,182),(24,154,46,181),(25,161,55,188),(26,168,56,187),(27,167,49,186),(28,166,50,185),(29,165,51,192),(30,164,52,191),(31,163,53,190),(32,162,54,189),(33,62,115,196),(34,61,116,195),(35,60,117,194),(36,59,118,193),(37,58,119,200),(38,57,120,199),(39,64,113,198),(40,63,114,197),(65,203,93,122),(66,202,94,121),(67,201,95,128),(68,208,96,127),(69,207,89,126),(70,206,90,125),(71,205,91,124),(72,204,92,123),(73,211,101,130),(74,210,102,129),(75,209,103,136),(76,216,104,135),(77,215,97,134),(78,214,98,133),(79,213,99,132),(80,212,100,131),(81,219,109,138),(82,218,110,137),(83,217,111,144),(84,224,112,143),(85,223,105,142),(86,222,106,141),(87,221,107,140),(88,220,108,139)], [(1,147,5,151),(2,146,6,150),(3,145,7,149),(4,152,8,148),(9,176,13,172),(10,175,14,171),(11,174,15,170),(12,173,16,169),(17,184,21,180),(18,183,22,179),(19,182,23,178),(20,181,24,177),(25,192,29,188),(26,191,30,187),(27,190,31,186),(28,189,32,185),(33,200,37,196),(34,199,38,195),(35,198,39,194),(36,197,40,193),(41,155,45,159),(42,154,46,158),(43,153,47,157),(44,160,48,156),(49,163,53,167),(50,162,54,166),(51,161,55,165),(52,168,56,164),(57,120,61,116),(58,119,62,115),(59,118,63,114),(60,117,64,113),(65,126,69,122),(66,125,70,121),(67,124,71,128),(68,123,72,127),(73,134,77,130),(74,133,78,129),(75,132,79,136),(76,131,80,135),(81,142,85,138),(82,141,86,137),(83,140,87,144),(84,139,88,143),(89,203,93,207),(90,202,94,206),(91,201,95,205),(92,208,96,204),(97,211,101,215),(98,210,102,214),(99,209,103,213),(100,216,104,212),(105,219,109,223),(106,218,110,222),(107,217,111,221),(108,224,112,220)]])

154 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A ··· 7F 8A ··· 8H 14A ··· 14R 14S ··· 14AD 28A ··· 28X 28Y ··· 28AV 56A ··· 56AV order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 ··· 7 8 ··· 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 2 2 2 2 2 2 8 8 8 8 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 8 ··· 8 2 ··· 2

154 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C7 C14 C14 C14 C14 C14 D4 D4 D4 C4○D4 Q16 C4○D8 C7×D4 C7×D4 C7×D4 C7×C4○D4 C7×Q16 C7×C4○D8 kernel C7×C8.18D4 C7×Q8⋊C4 C7×C2.D8 C7×C22⋊Q8 C22×C56 C14×Q16 C8.18D4 Q8⋊C4 C2.D8 C22⋊Q8 C22×C8 C2×Q16 C56 C2×C28 C22×C14 C28 C2×C14 C14 C8 C2×C4 C23 C4 C22 C2 # reps 1 2 1 2 1 1 6 12 6 12 6 6 2 1 1 2 4 4 12 6 6 12 24 24

Matrix representation of C7×C8.18D4 in GL4(𝔽113) generated by

 49 0 0 0 0 49 0 0 0 0 16 0 0 0 0 16
,
 98 0 0 0 0 15 0 0 0 0 95 0 0 0 0 69
,
 0 1 0 0 112 0 0 0 0 0 0 1 0 0 1 0
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 112 0
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,16,0,0,0,0,16],[98,0,0,0,0,15,0,0,0,0,95,0,0,0,0,69],[0,112,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[0,1,0,0,1,0,0,0,0,0,0,112,0,0,1,0] >;

C7×C8.18D4 in GAP, Magma, Sage, TeX

C_7\times C_8._{18}D_4
% in TeX

G:=Group("C7xC8.18D4");
// GroupNames label

G:=SmallGroup(448,875);
// by ID

G=gap.SmallGroup(448,875);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,1968,2438,9804,172]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^8=c^4=1,d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^4*c^-1>;
// generators/relations

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