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## G = C7×C42.C2order 224 = 25·7

### Direct product of C7 and C42.C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C7×C42.C2
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — C7×C42.C2
 Lower central C1 — C22 — C7×C42.C2
 Upper central C1 — C2×C14 — C7×C42.C2

Generators and relations for C7×C42.C2
G = < a,b,c,d | a7=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc2, dcd-1=b2c >

Subgroups: 68 in 56 conjugacy classes, 44 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C7, C2×C4, C2×C4, C14, C14, C42, C4⋊C4, C28, C28, C2×C14, C42.C2, C2×C28, C2×C28, C4×C28, C7×C4⋊C4, C7×C42.C2
Quotients: C1, C2, C22, C7, Q8, C23, C14, C2×Q8, C4○D4, C2×C14, C42.C2, C7×Q8, C22×C14, Q8×C14, C7×C4○D4, C7×C42.C2

Smallest permutation representation of C7×C42.C2
Regular action on 224 points
Generators in S224
(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 91 35 94)(2 85 29 95)(3 86 30 96)(4 87 31 97)(5 88 32 98)(6 89 33 92)(7 90 34 93)(8 186 18 168)(9 187 19 162)(10 188 20 163)(11 189 21 164)(12 183 15 165)(13 184 16 166)(14 185 17 167)(22 173 223 176)(23 174 224 177)(24 175 218 178)(25 169 219 179)(26 170 220 180)(27 171 221 181)(28 172 222 182)(36 100 44 82)(37 101 45 83)(38 102 46 84)(39 103 47 78)(40 104 48 79)(41 105 49 80)(42 99 43 81)(50 131 75 106)(51 132 76 107)(52 133 77 108)(53 127 71 109)(54 128 72 110)(55 129 73 111)(56 130 74 112)(57 123 67 113)(58 124 68 114)(59 125 69 115)(60 126 70 116)(61 120 64 117)(62 121 65 118)(63 122 66 119)(134 203 159 206)(135 197 160 207)(136 198 161 208)(137 199 155 209)(138 200 156 210)(139 201 157 204)(140 202 158 205)(141 191 151 216)(142 192 152 217)(143 193 153 211)(144 194 154 212)(145 195 148 213)(146 196 149 214)(147 190 150 215)
(1 50 39 66)(2 51 40 67)(3 52 41 68)(4 53 42 69)(5 54 36 70)(6 55 37 64)(7 56 38 65)(8 205 23 196)(9 206 24 190)(10 207 25 191)(11 208 26 192)(12 209 27 193)(13 210 28 194)(14 204 22 195)(15 199 221 211)(16 200 222 212)(17 201 223 213)(18 202 224 214)(19 203 218 215)(20 197 219 216)(21 198 220 217)(29 76 48 57)(30 77 49 58)(31 71 43 59)(32 72 44 60)(33 73 45 61)(34 74 46 62)(35 75 47 63)(78 122 94 106)(79 123 95 107)(80 124 96 108)(81 125 97 109)(82 126 98 110)(83 120 92 111)(84 121 93 112)(85 132 104 113)(86 133 105 114)(87 127 99 115)(88 128 100 116)(89 129 101 117)(90 130 102 118)(91 131 103 119)(134 175 150 187)(135 169 151 188)(136 170 152 189)(137 171 153 183)(138 172 154 184)(139 173 148 185)(140 174 149 186)(141 163 160 179)(142 164 161 180)(143 165 155 181)(144 166 156 182)(145 167 157 176)(146 168 158 177)(147 162 159 178)
(1 134 39 150)(2 135 40 151)(3 136 41 152)(4 137 42 153)(5 138 36 154)(6 139 37 148)(7 140 38 149)(8 118 23 130)(9 119 24 131)(10 113 25 132)(11 114 26 133)(12 115 27 127)(13 116 28 128)(14 117 22 129)(15 125 221 109)(16 126 222 110)(17 120 223 111)(18 121 224 112)(19 122 218 106)(20 123 219 107)(21 124 220 108)(29 160 48 141)(30 161 49 142)(31 155 43 143)(32 156 44 144)(33 157 45 145)(34 158 46 146)(35 159 47 147)(50 178 66 162)(51 179 67 163)(52 180 68 164)(53 181 69 165)(54 182 70 166)(55 176 64 167)(56 177 65 168)(57 188 76 169)(58 189 77 170)(59 183 71 171)(60 184 72 172)(61 185 73 173)(62 186 74 174)(63 187 75 175)(78 206 94 190)(79 207 95 191)(80 208 96 192)(81 209 97 193)(82 210 98 194)(83 204 92 195)(84 205 93 196)(85 216 104 197)(86 217 105 198)(87 211 99 199)(88 212 100 200)(89 213 101 201)(90 214 102 202)(91 215 103 203)

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

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

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

98 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4F 4G 4H 4I 4J 7A ··· 7F 14A ··· 14R 28A ··· 28AJ 28AK ··· 28BH order 1 2 2 2 4 ··· 4 4 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 1 ··· 1 2 ··· 2 4 ··· 4

98 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C2 C7 C14 C14 Q8 C4○D4 C7×Q8 C7×C4○D4 kernel C7×C42.C2 C4×C28 C7×C4⋊C4 C42.C2 C42 C4⋊C4 C28 C14 C4 C2 # reps 1 1 6 6 6 36 2 4 12 24

Matrix representation of C7×C42.C2 in GL4(𝔽29) generated by

 20 0 0 0 0 20 0 0 0 0 23 0 0 0 0 23
,
 1 0 0 0 0 28 0 0 0 0 17 0 0 0 0 17
,
 17 0 0 0 0 17 0 0 0 0 28 0 0 0 0 1
,
 0 1 0 0 28 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,28,0,0,0,0,17,0,0,0,0,17],[17,0,0,0,0,17,0,0,0,0,28,0,0,0,0,1],[0,28,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C7×C42.C2 in GAP, Magma, Sage, TeX

C_7\times C_4^2.C_2
% in TeX

G:=Group("C7xC4^2.C2");
// GroupNames label

G:=SmallGroup(224,160);
// by ID

G=gap.SmallGroup(224,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,697,679,2090,266]);
// Polycyclic

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

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