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G = C7×C42.6C4order 448 = 26·7

Direct product of C7 and C42.6C4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C42.6C4
 Chief series C1 — C2 — C4 — C2×C4 — C2×C28 — C2×C56 — C7×C22⋊C8 — C7×C42.6C4
 Lower central C1 — C22 — C7×C42.6C4
 Upper central C1 — C2×C28 — C7×C42.6C4

Generators and relations for C7×C42.6C4
G = < a,b,c,d | a7=b4=c4=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c2, dcd-1=b2c >

Subgroups: 146 in 110 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C2×C8, C22×C4, C28, C28, C2×C14, C2×C14, C2×C14, C8⋊C4, C22⋊C8, C4⋊C8, C2×C42, C56, C2×C28, C2×C28, C22×C14, C42.6C4, C4×C28, C2×C56, C22×C28, C7×C8⋊C4, C7×C22⋊C8, C7×C4⋊C8, C2×C4×C28, C7×C42.6C4
Quotients: C1, C2, C4, C22, C7, C2×C4, C23, C14, M4(2), C22×C4, C4○D4, C28, C2×C14, C42⋊C2, C2×M4(2), C2×C28, C22×C14, C42.6C4, C7×M4(2), C22×C28, C7×C4○D4, C7×C42⋊C2, C14×M4(2), C7×C42.6C4

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

196 conjugacy classes

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

196 irreducible representations

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

Matrix representation of C7×C42.6C4 in GL4(𝔽113) generated by

 49 0 0 0 0 49 0 0 0 0 28 0 0 0 0 28
,
 112 22 0 0 0 1 0 0 0 0 15 0 0 0 82 98
,
 98 0 0 0 0 98 0 0 0 0 1 0 0 0 13 112
,
 22 40 0 0 2 91 0 0 0 0 50 1 0 0 1 63
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,28,0,0,0,0,28],[112,0,0,0,22,1,0,0,0,0,15,82,0,0,0,98],[98,0,0,0,0,98,0,0,0,0,1,13,0,0,0,112],[22,2,0,0,40,91,0,0,0,0,50,1,0,0,1,63] >;

C7×C42.6C4 in GAP, Magma, Sage, TeX

C_7\times C_4^2._6C_4
% in TeX

G:=Group("C7xC4^2.6C4");
// GroupNames label

G:=SmallGroup(448,840);
// by ID

G=gap.SmallGroup(448,840);
# by ID

G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,310,124]);
// 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,d*c*d^-1=b^2*c>;
// generators/relations

׿
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