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## G = C2×C28.55D4order 448 = 26·7

### Direct product of C2 and C28.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C28.55D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C2×C7⋊C8 — C22×C7⋊C8 — C2×C28.55D4
 Lower central C7 — C14 — C2×C28.55D4
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C2×C28.55D4
G = < a,b,c,d | a2=b28=1, c4=b14, d2=b21, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b13, dcd-1=b7c3 >

Subgroups: 452 in 202 conjugacy classes, 103 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C2×C8, C22×C4, C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, C22×C8, C23×C4, C7⋊C8, C2×C28, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C2×C22⋊C8, C2×C7⋊C8, C2×C7⋊C8, C22×C28, C22×C28, C22×C28, C23×C14, C28.55D4, C22×C7⋊C8, C23×C28, C2×C28.55D4
Quotients:

Smallest permutation representation of C2×C28.55D4
On 224 points
Generators in S224
(1 174)(2 175)(3 176)(4 177)(5 178)(6 179)(7 180)(8 181)(9 182)(10 183)(11 184)(12 185)(13 186)(14 187)(15 188)(16 189)(17 190)(18 191)(19 192)(20 193)(21 194)(22 195)(23 196)(24 169)(25 170)(26 171)(27 172)(28 173)(29 152)(30 153)(31 154)(32 155)(33 156)(34 157)(35 158)(36 159)(37 160)(38 161)(39 162)(40 163)(41 164)(42 165)(43 166)(44 167)(45 168)(46 141)(47 142)(48 143)(49 144)(50 145)(51 146)(52 147)(53 148)(54 149)(55 150)(56 151)(57 203)(58 204)(59 205)(60 206)(61 207)(62 208)(63 209)(64 210)(65 211)(66 212)(67 213)(68 214)(69 215)(70 216)(71 217)(72 218)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 197)(80 198)(81 199)(82 200)(83 201)(84 202)(85 134)(86 135)(87 136)(88 137)(89 138)(90 139)(91 140)(92 113)(93 114)(94 115)(95 116)(96 117)(97 118)(98 119)(99 120)(100 121)(101 122)(102 123)(103 124)(104 125)(105 126)(106 127)(107 128)(108 129)(109 130)(110 131)(111 132)(112 133)
(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 29 202 90 15 43 216 104)(2 42 203 103 16 56 217 89)(3 55 204 88 17 41 218 102)(4 40 205 101 18 54 219 87)(5 53 206 86 19 39 220 100)(6 38 207 99 20 52 221 85)(7 51 208 112 21 37 222 98)(8 36 209 97 22 50 223 111)(9 49 210 110 23 35 224 96)(10 34 211 95 24 48 197 109)(11 47 212 108 25 33 198 94)(12 32 213 93 26 46 199 107)(13 45 214 106 27 31 200 92)(14 30 215 91 28 44 201 105)(57 124 189 151 71 138 175 165)(58 137 190 164 72 123 176 150)(59 122 191 149 73 136 177 163)(60 135 192 162 74 121 178 148)(61 120 193 147 75 134 179 161)(62 133 194 160 76 119 180 146)(63 118 195 145 77 132 181 159)(64 131 196 158 78 117 182 144)(65 116 169 143 79 130 183 157)(66 129 170 156 80 115 184 142)(67 114 171 141 81 128 185 155)(68 127 172 154 82 113 186 168)(69 140 173 167 83 126 187 153)(70 125 174 152 84 139 188 166)
(1 97 22 90 15 111 8 104)(2 110 23 103 16 96 9 89)(3 95 24 88 17 109 10 102)(4 108 25 101 18 94 11 87)(5 93 26 86 19 107 12 100)(6 106 27 99 20 92 13 85)(7 91 28 112 21 105 14 98)(29 202 50 223 43 216 36 209)(30 215 51 208 44 201 37 222)(31 200 52 221 45 214 38 207)(32 213 53 206 46 199 39 220)(33 198 54 219 47 212 40 205)(34 211 55 204 48 197 41 218)(35 224 56 217 49 210 42 203)(57 158 78 151 71 144 64 165)(58 143 79 164 72 157 65 150)(59 156 80 149 73 142 66 163)(60 141 81 162 74 155 67 148)(61 154 82 147 75 168 68 161)(62 167 83 160 76 153 69 146)(63 152 84 145 77 166 70 159)(113 186 134 179 127 172 120 193)(114 171 135 192 128 185 121 178)(115 184 136 177 129 170 122 191)(116 169 137 190 130 183 123 176)(117 182 138 175 131 196 124 189)(118 195 139 188 132 181 125 174)(119 180 140 173 133 194 126 187)

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

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

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

136 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 7A 7B 7C 8A ··· 8P 14A ··· 14AS 28A ··· 28AV order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 2 2 2 14 ··· 14 2 ··· 2 2 ··· 2

136 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 D4 D7 M4(2) Dic7 D14 Dic7 C7⋊D4 C7⋊C8 C4.Dic7 kernel C2×C28.55D4 C28.55D4 C22×C7⋊C8 C23×C28 C22×C28 C23×C14 C22×C14 C2×C28 C23×C4 C2×C14 C22×C4 C22×C4 C24 C2×C4 C23 C22 # reps 1 4 2 1 6 2 16 4 3 4 9 9 3 24 24 24

Matrix representation of C2×C28.55D4 in GL4(𝔽113) generated by

 1 0 0 0 0 112 0 0 0 0 1 0 0 0 0 1
,
 98 0 0 0 0 112 0 0 0 0 30 1 0 0 0 49
,
 95 0 0 0 0 98 0 0 0 0 53 55 0 0 103 60
,
 18 0 0 0 0 15 0 0 0 0 60 13 0 0 10 53
G:=sub<GL(4,GF(113))| [1,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[98,0,0,0,0,112,0,0,0,0,30,0,0,0,1,49],[95,0,0,0,0,98,0,0,0,0,53,103,0,0,55,60],[18,0,0,0,0,15,0,0,0,0,60,10,0,0,13,53] >;

C2×C28.55D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}._{55}D_4
% in TeX

G:=Group("C2xC28.55D4");
// GroupNames label

G:=SmallGroup(448,740);
// by ID

G=gap.SmallGroup(448,740);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,136,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^28=1,c^4=b^14,d^2=b^21,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^13,d*c*d^-1=b^7*c^3>;
// generators/relations

׿
×
𝔽