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## G = C22×C7⋊C8order 224 = 25·7

### Direct product of C22 and C7⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×C7⋊C8
 Chief series C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8 — C22×C7⋊C8
 Lower central C7 — C22×C7⋊C8
 Upper central C1 — C22×C4

Generators and relations for C22×C7⋊C8
G = < a,b,c,d | a2=b2=c7=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 142 in 76 conjugacy classes, 65 normal (11 characteristic)
C1, C2, C2, C4, C4, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, C22×C4, C28, C28, C2×C14, C22×C8, C7⋊C8, C2×C28, C22×C14, C2×C7⋊C8, C22×C28, C22×C7⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, D7, C2×C8, C22×C4, Dic7, D14, C22×C8, C7⋊C8, C2×Dic7, C22×D7, C2×C7⋊C8, C22×Dic7, C22×C7⋊C8

Smallest permutation representation of C22×C7⋊C8
Regular action on 224 points
Generators in S224
(1 161)(2 162)(3 163)(4 164)(5 165)(6 166)(7 167)(8 168)(9 152)(10 145)(11 146)(12 147)(13 148)(14 149)(15 150)(16 151)(17 160)(18 153)(19 154)(20 155)(21 156)(22 157)(23 158)(24 159)(25 144)(26 137)(27 138)(28 139)(29 140)(30 141)(31 142)(32 143)(33 92)(34 93)(35 94)(36 95)(37 96)(38 89)(39 90)(40 91)(41 100)(42 101)(43 102)(44 103)(45 104)(46 97)(47 98)(48 99)(49 108)(50 109)(51 110)(52 111)(53 112)(54 105)(55 106)(56 107)(57 196)(58 197)(59 198)(60 199)(61 200)(62 193)(63 194)(64 195)(65 204)(66 205)(67 206)(68 207)(69 208)(70 201)(71 202)(72 203)(73 212)(74 213)(75 214)(76 215)(77 216)(78 209)(79 210)(80 211)(81 220)(82 221)(83 222)(84 223)(85 224)(86 217)(87 218)(88 219)(113 174)(114 175)(115 176)(116 169)(117 170)(118 171)(119 172)(120 173)(121 182)(122 183)(123 184)(124 177)(125 178)(126 179)(127 180)(128 181)(129 190)(130 191)(131 192)(132 185)(133 186)(134 187)(135 188)(136 189)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 91)(10 92)(11 93)(12 94)(13 95)(14 96)(15 89)(16 90)(17 106)(18 107)(19 108)(20 109)(21 110)(22 111)(23 112)(24 105)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 69)(42 70)(43 71)(44 72)(45 65)(46 66)(47 67)(48 68)(49 154)(50 155)(51 156)(52 157)(53 158)(54 159)(55 160)(56 153)(57 170)(58 171)(59 172)(60 173)(61 174)(62 175)(63 176)(64 169)(73 179)(74 180)(75 181)(76 182)(77 183)(78 184)(79 177)(80 178)(81 186)(82 187)(83 188)(84 189)(85 190)(86 191)(87 192)(88 185)(97 205)(98 206)(99 207)(100 208)(101 201)(102 202)(103 203)(104 204)(113 200)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 199)(121 215)(122 216)(123 209)(124 210)(125 211)(126 212)(127 213)(128 214)(129 224)(130 217)(131 218)(132 219)(133 220)(134 221)(135 222)(136 223)(137 166)(138 167)(139 168)(140 161)(141 162)(142 163)(143 164)(144 165)
(1 117 92 205 216 17 217)(2 218 18 209 206 93 118)(3 119 94 207 210 19 219)(4 220 20 211 208 95 120)(5 113 96 201 212 21 221)(6 222 22 213 202 89 114)(7 115 90 203 214 23 223)(8 224 24 215 204 91 116)(9 195 28 129 105 121 104)(10 97 122 106 130 29 196)(11 197 30 131 107 123 98)(12 99 124 108 132 31 198)(13 199 32 133 109 125 100)(14 101 126 110 134 25 200)(15 193 26 135 111 127 102)(16 103 128 112 136 27 194)(33 66 77 160 86 161 170)(34 171 162 87 153 78 67)(35 68 79 154 88 163 172)(36 173 164 81 155 80 69)(37 70 73 156 82 165 174)(38 175 166 83 157 74 71)(39 72 75 158 84 167 176)(40 169 168 85 159 76 65)(41 148 60 143 186 50 178)(42 179 51 187 144 61 149)(43 150 62 137 188 52 180)(44 181 53 189 138 63 151)(45 152 64 139 190 54 182)(46 183 55 191 140 57 145)(47 146 58 141 192 56 184)(48 177 49 185 142 59 147)
(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)| (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,152)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,160)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,144)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,92)(34,93)(35,94)(36,95)(37,96)(38,89)(39,90)(40,91)(41,100)(42,101)(43,102)(44,103)(45,104)(46,97)(47,98)(48,99)(49,108)(50,109)(51,110)(52,111)(53,112)(54,105)(55,106)(56,107)(57,196)(58,197)(59,198)(60,199)(61,200)(62,193)(63,194)(64,195)(65,204)(66,205)(67,206)(68,207)(69,208)(70,201)(71,202)(72,203)(73,212)(74,213)(75,214)(76,215)(77,216)(78,209)(79,210)(80,211)(81,220)(82,221)(83,222)(84,223)(85,224)(86,217)(87,218)(88,219)(113,174)(114,175)(115,176)(116,169)(117,170)(118,171)(119,172)(120,173)(121,182)(122,183)(123,184)(124,177)(125,178)(126,179)(127,180)(128,181)(129,190)(130,191)(131,192)(132,185)(133,186)(134,187)(135,188)(136,189), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,105)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,69)(42,70)(43,71)(44,72)(45,65)(46,66)(47,67)(48,68)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,153)(57,170)(58,171)(59,172)(60,173)(61,174)(62,175)(63,176)(64,169)(73,179)(74,180)(75,181)(76,182)(77,183)(78,184)(79,177)(80,178)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,185)(97,205)(98,206)(99,207)(100,208)(101,201)(102,202)(103,203)(104,204)(113,200)(114,193)(115,194)(116,195)(117,196)(118,197)(119,198)(120,199)(121,215)(122,216)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,224)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,166)(138,167)(139,168)(140,161)(141,162)(142,163)(143,164)(144,165), (1,117,92,205,216,17,217)(2,218,18,209,206,93,118)(3,119,94,207,210,19,219)(4,220,20,211,208,95,120)(5,113,96,201,212,21,221)(6,222,22,213,202,89,114)(7,115,90,203,214,23,223)(8,224,24,215,204,91,116)(9,195,28,129,105,121,104)(10,97,122,106,130,29,196)(11,197,30,131,107,123,98)(12,99,124,108,132,31,198)(13,199,32,133,109,125,100)(14,101,126,110,134,25,200)(15,193,26,135,111,127,102)(16,103,128,112,136,27,194)(33,66,77,160,86,161,170)(34,171,162,87,153,78,67)(35,68,79,154,88,163,172)(36,173,164,81,155,80,69)(37,70,73,156,82,165,174)(38,175,166,83,157,74,71)(39,72,75,158,84,167,176)(40,169,168,85,159,76,65)(41,148,60,143,186,50,178)(42,179,51,187,144,61,149)(43,150,62,137,188,52,180)(44,181,53,189,138,63,151)(45,152,64,139,190,54,182)(46,183,55,191,140,57,145)(47,146,58,141,192,56,184)(48,177,49,185,142,59,147), (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( (1,161)(2,162)(3,163)(4,164)(5,165)(6,166)(7,167)(8,168)(9,152)(10,145)(11,146)(12,147)(13,148)(14,149)(15,150)(16,151)(17,160)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,144)(26,137)(27,138)(28,139)(29,140)(30,141)(31,142)(32,143)(33,92)(34,93)(35,94)(36,95)(37,96)(38,89)(39,90)(40,91)(41,100)(42,101)(43,102)(44,103)(45,104)(46,97)(47,98)(48,99)(49,108)(50,109)(51,110)(52,111)(53,112)(54,105)(55,106)(56,107)(57,196)(58,197)(59,198)(60,199)(61,200)(62,193)(63,194)(64,195)(65,204)(66,205)(67,206)(68,207)(69,208)(70,201)(71,202)(72,203)(73,212)(74,213)(75,214)(76,215)(77,216)(78,209)(79,210)(80,211)(81,220)(82,221)(83,222)(84,223)(85,224)(86,217)(87,218)(88,219)(113,174)(114,175)(115,176)(116,169)(117,170)(118,171)(119,172)(120,173)(121,182)(122,183)(123,184)(124,177)(125,178)(126,179)(127,180)(128,181)(129,190)(130,191)(131,192)(132,185)(133,186)(134,187)(135,188)(136,189), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,91)(10,92)(11,93)(12,94)(13,95)(14,96)(15,89)(16,90)(17,106)(18,107)(19,108)(20,109)(21,110)(22,111)(23,112)(24,105)(33,145)(34,146)(35,147)(36,148)(37,149)(38,150)(39,151)(40,152)(41,69)(42,70)(43,71)(44,72)(45,65)(46,66)(47,67)(48,68)(49,154)(50,155)(51,156)(52,157)(53,158)(54,159)(55,160)(56,153)(57,170)(58,171)(59,172)(60,173)(61,174)(62,175)(63,176)(64,169)(73,179)(74,180)(75,181)(76,182)(77,183)(78,184)(79,177)(80,178)(81,186)(82,187)(83,188)(84,189)(85,190)(86,191)(87,192)(88,185)(97,205)(98,206)(99,207)(100,208)(101,201)(102,202)(103,203)(104,204)(113,200)(114,193)(115,194)(116,195)(117,196)(118,197)(119,198)(120,199)(121,215)(122,216)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,224)(130,217)(131,218)(132,219)(133,220)(134,221)(135,222)(136,223)(137,166)(138,167)(139,168)(140,161)(141,162)(142,163)(143,164)(144,165), (1,117,92,205,216,17,217)(2,218,18,209,206,93,118)(3,119,94,207,210,19,219)(4,220,20,211,208,95,120)(5,113,96,201,212,21,221)(6,222,22,213,202,89,114)(7,115,90,203,214,23,223)(8,224,24,215,204,91,116)(9,195,28,129,105,121,104)(10,97,122,106,130,29,196)(11,197,30,131,107,123,98)(12,99,124,108,132,31,198)(13,199,32,133,109,125,100)(14,101,126,110,134,25,200)(15,193,26,135,111,127,102)(16,103,128,112,136,27,194)(33,66,77,160,86,161,170)(34,171,162,87,153,78,67)(35,68,79,154,88,163,172)(36,173,164,81,155,80,69)(37,70,73,156,82,165,174)(38,175,166,83,157,74,71)(39,72,75,158,84,167,176)(40,169,168,85,159,76,65)(41,148,60,143,186,50,178)(42,179,51,187,144,61,149)(43,150,62,137,188,52,180)(44,181,53,189,138,63,151)(45,152,64,139,190,54,182)(46,183,55,191,140,57,145)(47,146,58,141,192,56,184)(48,177,49,185,142,59,147), (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([[(1,161),(2,162),(3,163),(4,164),(5,165),(6,166),(7,167),(8,168),(9,152),(10,145),(11,146),(12,147),(13,148),(14,149),(15,150),(16,151),(17,160),(18,153),(19,154),(20,155),(21,156),(22,157),(23,158),(24,159),(25,144),(26,137),(27,138),(28,139),(29,140),(30,141),(31,142),(32,143),(33,92),(34,93),(35,94),(36,95),(37,96),(38,89),(39,90),(40,91),(41,100),(42,101),(43,102),(44,103),(45,104),(46,97),(47,98),(48,99),(49,108),(50,109),(51,110),(52,111),(53,112),(54,105),(55,106),(56,107),(57,196),(58,197),(59,198),(60,199),(61,200),(62,193),(63,194),(64,195),(65,204),(66,205),(67,206),(68,207),(69,208),(70,201),(71,202),(72,203),(73,212),(74,213),(75,214),(76,215),(77,216),(78,209),(79,210),(80,211),(81,220),(82,221),(83,222),(84,223),(85,224),(86,217),(87,218),(88,219),(113,174),(114,175),(115,176),(116,169),(117,170),(118,171),(119,172),(120,173),(121,182),(122,183),(123,184),(124,177),(125,178),(126,179),(127,180),(128,181),(129,190),(130,191),(131,192),(132,185),(133,186),(134,187),(135,188),(136,189)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,91),(10,92),(11,93),(12,94),(13,95),(14,96),(15,89),(16,90),(17,106),(18,107),(19,108),(20,109),(21,110),(22,111),(23,112),(24,105),(33,145),(34,146),(35,147),(36,148),(37,149),(38,150),(39,151),(40,152),(41,69),(42,70),(43,71),(44,72),(45,65),(46,66),(47,67),(48,68),(49,154),(50,155),(51,156),(52,157),(53,158),(54,159),(55,160),(56,153),(57,170),(58,171),(59,172),(60,173),(61,174),(62,175),(63,176),(64,169),(73,179),(74,180),(75,181),(76,182),(77,183),(78,184),(79,177),(80,178),(81,186),(82,187),(83,188),(84,189),(85,190),(86,191),(87,192),(88,185),(97,205),(98,206),(99,207),(100,208),(101,201),(102,202),(103,203),(104,204),(113,200),(114,193),(115,194),(116,195),(117,196),(118,197),(119,198),(120,199),(121,215),(122,216),(123,209),(124,210),(125,211),(126,212),(127,213),(128,214),(129,224),(130,217),(131,218),(132,219),(133,220),(134,221),(135,222),(136,223),(137,166),(138,167),(139,168),(140,161),(141,162),(142,163),(143,164),(144,165)], [(1,117,92,205,216,17,217),(2,218,18,209,206,93,118),(3,119,94,207,210,19,219),(4,220,20,211,208,95,120),(5,113,96,201,212,21,221),(6,222,22,213,202,89,114),(7,115,90,203,214,23,223),(8,224,24,215,204,91,116),(9,195,28,129,105,121,104),(10,97,122,106,130,29,196),(11,197,30,131,107,123,98),(12,99,124,108,132,31,198),(13,199,32,133,109,125,100),(14,101,126,110,134,25,200),(15,193,26,135,111,127,102),(16,103,128,112,136,27,194),(33,66,77,160,86,161,170),(34,171,162,87,153,78,67),(35,68,79,154,88,163,172),(36,173,164,81,155,80,69),(37,70,73,156,82,165,174),(38,175,166,83,157,74,71),(39,72,75,158,84,167,176),(40,169,168,85,159,76,65),(41,148,60,143,186,50,178),(42,179,51,187,144,61,149),(43,150,62,137,188,52,180),(44,181,53,189,138,63,151),(45,152,64,139,190,54,182),(46,183,55,191,140,57,145),(47,146,58,141,192,56,184),(48,177,49,185,142,59,147)], [(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)]])

C22×C7⋊C8 is a maximal subgroup of
(C2×C28)⋊3C8  C28.C42  (C2×C56)⋊5C4  C28.4C42  C7⋊D4⋊C8  C7⋊C826D4  C28.5C42  C28.45(C4⋊C4)  C42.43D14  C4.(C2×D28)  C42.47D14  C7⋊C822D4  C7⋊C823D4  C7⋊C824D4  C7⋊C8.29D4  C2×C8×Dic7  C28.439(C2×D4)  C28.7C42  (C2×D28).14C4  C28.(C2×D4)  (D4×C14).11C4  D7×C22×C8
C22×C7⋊C8 is a maximal quotient of
C42.6Dic7  C56.70C23

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 7A 7B 7C 8A ··· 8P 14A ··· 14U 28A ··· 28X order 1 2 ··· 2 4 ··· 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 ··· 1 1 ··· 1 2 2 2 7 ··· 7 2 ··· 2 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 D7 Dic7 D14 Dic7 C7⋊C8 kernel C22×C7⋊C8 C2×C7⋊C8 C22×C28 C2×C28 C22×C14 C2×C14 C22×C4 C2×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 16 3 9 9 3 24

Matrix representation of C22×C7⋊C8 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 9 112 0 0 1 0
,
 69 0 0 0 0 1 0 0 0 0 82 56 0 0 3 31
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,112,0],[69,0,0,0,0,1,0,0,0,0,82,3,0,0,56,31] >;

C22×C7⋊C8 in GAP, Magma, Sage, TeX

C_2^2\times C_7\rtimes C_8
% in TeX

G:=Group("C2^2xC7:C8");
// GroupNames label

G:=SmallGroup(224,115);
// by ID

G=gap.SmallGroup(224,115);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,48,69,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^7=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽