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## G = C7×C42.12C4order 448 = 26·7

### Direct product of C7 and C42.12C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C42.12C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C2×C56 — C7×C22⋊C8 — C7×C42.12C4
 Lower central C1 — C2 — C7×C42.12C4
 Upper central C1 — C4×C28 — C7×C42.12C4

Generators and relations for C7×C42.12C4
G = < a,b,c,d | a7=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, cd=dc >

Subgroups: 146 in 118 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C56, C2×C28, C2×C28, C2×C28, C22×C14, C42.12C4, C4×C28, C2×C56, C22×C28, C4×C56, C7×C22⋊C8, C7×C4⋊C8, C2×C4×C28, C7×C42.12C4
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C2×C8, M4(2), C22×C4, C4○D4, C28, C2×C14, C42⋊C2, C22×C8, C2×M4(2), C56, C2×C28, C22×C14, C42.12C4, C2×C56, C7×M4(2), C22×C28, C7×C4○D4, C7×C42⋊C2, C22×C56, C14×M4(2), C7×C42.12C4

Smallest permutation representation of C7×C42.12C4
On 224 points
Generators in S224
(1 151 39 143 31 135 23)(2 152 40 144 32 136 24)(3 145 33 137 25 129 17)(4 146 34 138 26 130 18)(5 147 35 139 27 131 19)(6 148 36 140 28 132 20)(7 149 37 141 29 133 21)(8 150 38 142 30 134 22)(9 122 58 162 50 154 42)(10 123 59 163 51 155 43)(11 124 60 164 52 156 44)(12 125 61 165 53 157 45)(13 126 62 166 54 158 46)(14 127 63 167 55 159 47)(15 128 64 168 56 160 48)(16 121 57 161 49 153 41)(65 169 193 81 185 73 177)(66 170 194 82 186 74 178)(67 171 195 83 187 75 179)(68 172 196 84 188 76 180)(69 173 197 85 189 77 181)(70 174 198 86 190 78 182)(71 175 199 87 191 79 183)(72 176 200 88 192 80 184)(89 117 217 105 209 97 201)(90 118 218 106 210 98 202)(91 119 219 107 211 99 203)(92 120 220 108 212 100 204)(93 113 221 109 213 101 205)(94 114 222 110 214 102 206)(95 115 223 111 215 103 207)(96 116 224 112 216 104 208)
(1 7 5 3)(2 174 6 170)(4 176 8 172)(9 116 13 120)(10 16 14 12)(11 118 15 114)(17 23 21 19)(18 72 22 68)(20 66 24 70)(25 31 29 27)(26 80 30 76)(28 74 32 78)(33 39 37 35)(34 88 38 84)(36 82 40 86)(41 47 45 43)(42 96 46 92)(44 90 48 94)(49 55 53 51)(50 104 54 100)(52 98 56 102)(57 63 61 59)(58 112 62 108)(60 106 64 110)(65 71 69 67)(73 79 77 75)(81 87 85 83)(89 95 93 91)(97 103 101 99)(105 111 109 107)(113 119 117 115)(121 127 125 123)(122 224 126 220)(124 218 128 222)(129 135 133 131)(130 184 134 180)(132 178 136 182)(137 143 141 139)(138 192 142 188)(140 186 144 190)(145 151 149 147)(146 200 150 196)(148 194 152 198)(153 159 157 155)(154 208 158 204)(156 202 160 206)(161 167 165 163)(162 216 166 212)(164 210 168 214)(169 175 173 171)(177 183 181 179)(185 191 189 187)(193 199 197 195)(201 207 205 203)(209 215 213 211)(217 223 221 219)
(1 89 171 45)(2 90 172 46)(3 91 173 47)(4 92 174 48)(5 93 175 41)(6 94 176 42)(7 95 169 43)(8 96 170 44)(9 148 114 200)(10 149 115 193)(11 150 116 194)(12 151 117 195)(13 152 118 196)(14 145 119 197)(15 146 120 198)(16 147 113 199)(17 203 69 159)(18 204 70 160)(19 205 71 153)(20 206 72 154)(21 207 65 155)(22 208 66 156)(23 201 67 157)(24 202 68 158)(25 211 77 167)(26 212 78 168)(27 213 79 161)(28 214 80 162)(29 215 73 163)(30 216 74 164)(31 209 75 165)(32 210 76 166)(33 219 85 127)(34 220 86 128)(35 221 87 121)(36 222 88 122)(37 223 81 123)(38 224 82 124)(39 217 83 125)(40 218 84 126)(49 131 101 183)(50 132 102 184)(51 133 103 177)(52 134 104 178)(53 135 97 179)(54 136 98 180)(55 129 99 181)(56 130 100 182)(57 139 109 191)(58 140 110 192)(59 141 111 185)(60 142 112 186)(61 143 105 187)(62 144 106 188)(63 137 107 189)(64 138 108 190)
(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)

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

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

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

280 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 7A ··· 7F 8A ··· 8P 14A ··· 14R 14S ··· 14AD 28A ··· 28BT 28BU ··· 28DD 56A ··· 56CR order 1 2 2 2 2 2 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 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

280 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C7 C8 C14 C14 C14 C14 C28 C28 C56 M4(2) C4○D4 C7×M4(2) C7×C4○D4 kernel C7×C42.12C4 C4×C56 C7×C22⋊C8 C7×C4⋊C8 C2×C4×C28 C4×C28 C22×C28 C42.12C4 C2×C28 C4×C8 C22⋊C8 C4⋊C8 C2×C42 C42 C22×C4 C2×C4 C28 C28 C4 C4 # reps 1 2 2 2 1 4 4 6 16 12 12 12 6 24 24 96 4 4 24 24

Matrix representation of C7×C42.12C4 in GL3(𝔽113) generated by

 1 0 0 0 30 0 0 0 30
,
 15 0 0 0 98 0 0 25 15
,
 98 0 0 0 1 0 0 0 1
,
 69 0 0 0 69 15 0 0 44
G:=sub<GL(3,GF(113))| [1,0,0,0,30,0,0,0,30],[15,0,0,0,98,25,0,0,15],[98,0,0,0,1,0,0,0,1],[69,0,0,0,69,0,0,15,44] >;

C7×C42.12C4 in GAP, Magma, Sage, TeX

C_7\times C_4^2._{12}C_4
% in TeX

G:=Group("C7xC4^2.12C4");
// GroupNames label

G:=SmallGroup(448,839);
// by ID

G=gap.SmallGroup(448,839);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,310,124]);
// Polycyclic

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

׿
×
𝔽