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## G = C4×C4.Dic7order 448 = 26·7

### Direct product of C4 and C4.Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C4×C4.Dic7
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C2×C4.Dic7 — C4×C4.Dic7
 Lower central C7 — C14 — C4×C4.Dic7
 Upper central C1 — C42 — C2×C42

Generators and relations for C4×C4.Dic7
G = < a,b,c,d | a4=b4=1, c14=b2, d2=b2c7, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c13 >

Subgroups: 324 in 142 conjugacy classes, 95 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C4×M4(2), C2×C7⋊C8, C4.Dic7, C4×C28, C4×C28, C22×C28, C22×C28, C4×C7⋊C8, C42.D7, C2×C4.Dic7, C2×C4×C28, C4×C4.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C42, M4(2), C22×C4, Dic7, D14, C2×C42, C2×M4(2), C4×D7, C2×Dic7, C22×D7, C4×M4(2), C4.Dic7, C4×Dic7, C2×C4×D7, C22×Dic7, C2×C4.Dic7, C2×C4×Dic7, C4×C4.Dic7

Smallest permutation representation of C4×C4.Dic7
On 224 points
Generators in S224
(1 148 222 71)(2 149 223 72)(3 150 224 73)(4 151 197 74)(5 152 198 75)(6 153 199 76)(7 154 200 77)(8 155 201 78)(9 156 202 79)(10 157 203 80)(11 158 204 81)(12 159 205 82)(13 160 206 83)(14 161 207 84)(15 162 208 57)(16 163 209 58)(17 164 210 59)(18 165 211 60)(19 166 212 61)(20 167 213 62)(21 168 214 63)(22 141 215 64)(23 142 216 65)(24 143 217 66)(25 144 218 67)(26 145 219 68)(27 146 220 69)(28 147 221 70)(29 113 170 87)(30 114 171 88)(31 115 172 89)(32 116 173 90)(33 117 174 91)(34 118 175 92)(35 119 176 93)(36 120 177 94)(37 121 178 95)(38 122 179 96)(39 123 180 97)(40 124 181 98)(41 125 182 99)(42 126 183 100)(43 127 184 101)(44 128 185 102)(45 129 186 103)(46 130 187 104)(47 131 188 105)(48 132 189 106)(49 133 190 107)(50 134 191 108)(51 135 192 109)(52 136 193 110)(53 137 194 111)(54 138 195 112)(55 139 196 85)(56 140 169 86)
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 36 43 50)(30 37 44 51)(31 38 45 52)(32 39 46 53)(33 40 47 54)(34 41 48 55)(35 42 49 56)(57 78 71 64)(58 79 72 65)(59 80 73 66)(60 81 74 67)(61 82 75 68)(62 83 76 69)(63 84 77 70)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(113 120 127 134)(114 121 128 135)(115 122 129 136)(116 123 130 137)(117 124 131 138)(118 125 132 139)(119 126 133 140)(141 162 155 148)(142 163 156 149)(143 164 157 150)(144 165 158 151)(145 166 159 152)(146 167 160 153)(147 168 161 154)(169 176 183 190)(170 177 184 191)(171 178 185 192)(172 179 186 193)(173 180 187 194)(174 181 188 195)(175 182 189 196)(197 218 211 204)(198 219 212 205)(199 220 213 206)(200 221 214 207)(201 222 215 208)(202 223 216 209)(203 224 217 210)
(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)
(1 106 22 99 15 92 8 85)(2 91 23 112 16 105 9 98)(3 104 24 97 17 90 10 111)(4 89 25 110 18 103 11 96)(5 102 26 95 19 88 12 109)(6 87 27 108 20 101 13 94)(7 100 28 93 21 86 14 107)(29 146 50 167 43 160 36 153)(30 159 51 152 44 145 37 166)(31 144 52 165 45 158 38 151)(32 157 53 150 46 143 39 164)(33 142 54 163 47 156 40 149)(34 155 55 148 48 141 41 162)(35 168 56 161 49 154 42 147)(57 175 78 196 71 189 64 182)(58 188 79 181 72 174 65 195)(59 173 80 194 73 187 66 180)(60 186 81 179 74 172 67 193)(61 171 82 192 75 185 68 178)(62 184 83 177 76 170 69 191)(63 169 84 190 77 183 70 176)(113 220 134 213 127 206 120 199)(114 205 135 198 128 219 121 212)(115 218 136 211 129 204 122 197)(116 203 137 224 130 217 123 210)(117 216 138 209 131 202 124 223)(118 201 139 222 132 215 125 208)(119 214 140 207 133 200 126 221)

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

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

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

136 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 7A 7B 7C 8A ··· 8P 14A ··· 14U 28A ··· 28BT order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 C4 D7 M4(2) Dic7 D14 Dic7 D14 C4×D7 C4.Dic7 kernel C4×C4.Dic7 C4×C7⋊C8 C42.D7 C2×C4.Dic7 C2×C4×C28 C4.Dic7 C4×C28 C22×C28 C2×C42 C28 C42 C42 C22×C4 C22×C4 C2×C4 C4 # reps 1 2 2 2 1 16 4 4 3 8 6 6 6 3 24 48

Matrix representation of C4×C4.Dic7 in GL3(𝔽113) generated by

 98 0 0 0 98 0 0 0 98
,
 1 0 0 0 98 0 0 0 15
,
 112 0 0 0 60 0 0 0 32
,
 98 0 0 0 0 112 0 15 0
G:=sub<GL(3,GF(113))| [98,0,0,0,98,0,0,0,98],[1,0,0,0,98,0,0,0,15],[112,0,0,0,60,0,0,0,32],[98,0,0,0,0,15,0,112,0] >;

C4×C4.Dic7 in GAP, Magma, Sage, TeX

C_4\times C_4.{\rm Dic}_7
% in TeX

G:=Group("C4xC4.Dic7");
// GroupNames label

G:=SmallGroup(448,456);
// by ID

G=gap.SmallGroup(448,456);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,477,100,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^14=b^2,d^2=b^2*c^7,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^13>;
// generators/relations

׿
×
𝔽