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## G = C2×C28.10D4order 448 = 26·7

### Direct product of C2 and C28.10D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C28.10D4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4.Dic7 — C2×C4.Dic7 — C2×C28.10D4
 Lower central C7 — C14 — C2×C14 — C2×C28.10D4
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for C2×C28.10D4
G = < a,b,c,d | a2=b28=1, c4=b14, d2=b21, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=b13, dcd-1=b21c3 >

Subgroups: 372 in 146 conjugacy classes, 71 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, Q8, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C2×Q8, C2×Q8, C28, C28, C2×C14, C2×C14, C4.10D4, C2×M4(2), C22×Q8, C7⋊C8, C2×C28, C2×C28, C2×C28, C7×Q8, C22×C14, C2×C4.10D4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C22×C28, C22×C28, Q8×C14, Q8×C14, C28.10D4, C2×C4.Dic7, Q8×C2×C14, C2×C28.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C4.10D4, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C2×C4.10D4, C23.D7, C22×Dic7, C2×C7⋊D4, C28.10D4, C2×C23.D7, C2×C28.10D4

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

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 7A 7B 7C 8A ··· 8H 14A ··· 14U 28A ··· 28AJ order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 2 2 2 28 ··· 28 2 ··· 2 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + - + - + - image C1 C2 C2 C2 C4 C4 D4 D7 Dic7 D14 Dic7 D14 C7⋊D4 C4.10D4 C28.10D4 kernel C2×C28.10D4 C28.10D4 C2×C4.Dic7 Q8×C2×C14 C22×C28 Q8×C14 C2×C28 C22×Q8 C22×C4 C22×C4 C2×Q8 C2×Q8 C2×C4 C14 C2 # reps 1 4 2 1 4 4 4 3 6 3 6 6 24 2 12

Matrix representation of C2×C28.10D4 in GL6(𝔽113)

 112 0 0 0 0 0 0 112 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 103 24 0 0 0 0 89 1 0 0 0 0 0 0 32 72 0 0 0 0 25 81 0 0 0 0 83 20 0 1 0 0 8 88 112 0
,
 9 105 0 0 0 0 95 104 0 0 0 0 0 0 93 0 72 0 0 0 98 0 81 112 0 0 107 112 20 0 0 0 65 0 88 0
,
 9 105 0 0 0 0 95 104 0 0 0 0 0 0 60 0 37 112 0 0 19 0 95 81 0 0 7 45 4 6 0 0 98 11 108 49

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[103,89,0,0,0,0,24,1,0,0,0,0,0,0,32,25,83,8,0,0,72,81,20,88,0,0,0,0,0,112,0,0,0,0,1,0],[9,95,0,0,0,0,105,104,0,0,0,0,0,0,93,98,107,65,0,0,0,0,112,0,0,0,72,81,20,88,0,0,0,112,0,0],[9,95,0,0,0,0,105,104,0,0,0,0,0,0,60,19,7,98,0,0,0,0,45,11,0,0,37,95,4,108,0,0,112,81,6,49] >;

C2×C28.10D4 in GAP, Magma, Sage, TeX

C_2\times C_{28}._{10}D_4
% in TeX

G:=Group("C2xC28.10D4");
// GroupNames label

G:=SmallGroup(448,760);
// by ID

G=gap.SmallGroup(448,760);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,184,297,136,1684,18822]);
// 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=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^21*c^3>;
// generators/relations

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