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## G = C7×C2.C42order 224 = 25·7

### Direct product of C7 and C2.C42

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C7×C2.C42
 Chief series C1 — C2 — C22 — C23 — C22×C14 — C22×C28 — C7×C2.C42
 Lower central C1 — C2 — C7×C2.C42
 Upper central C1 — C22×C14 — C7×C2.C42

Generators and relations for C7×C2.C42
G = < a,b,c,d | a7=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >

Subgroups: 100 in 76 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, C14, C14, C22×C4, C28, C2×C14, C2×C14, C2.C42, C2×C28, C2×C28, C22×C14, C22×C28, C7×C2.C42
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C2×C28, C7×D4, C7×Q8, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42

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

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

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

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

140 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4L 7A ··· 7F 14A ··· 14AP 28A ··· 28BT order 1 2 ··· 2 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + - image C1 C2 C4 C7 C14 C28 D4 Q8 C7×D4 C7×Q8 kernel C7×C2.C42 C22×C28 C2×C28 C2.C42 C22×C4 C2×C4 C2×C14 C2×C14 C22 C22 # reps 1 3 12 6 18 72 3 1 18 6

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

 1 0 0 0 0 1 0 0 0 0 25 0 0 0 0 25
,
 1 0 0 0 0 1 0 0 0 0 28 0 0 0 0 28
,
 12 0 0 0 0 17 0 0 0 0 6 28 0 0 8 23
,
 1 0 0 0 0 12 0 0 0 0 28 27 0 0 1 1
G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,25,0,0,0,0,25],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[12,0,0,0,0,17,0,0,0,0,6,8,0,0,28,23],[1,0,0,0,0,12,0,0,0,0,28,1,0,0,27,1] >;

C7×C2.C42 in GAP, Magma, Sage, TeX

C_7\times C_2.C_4^2
% in TeX

G:=Group("C7xC2.C4^2");
// GroupNames label

G:=SmallGroup(224,44);
// by ID

G=gap.SmallGroup(224,44);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,336,361,679]);
// Polycyclic

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

׿
×
𝔽