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## G = C2×Q8.D14order 448 = 26·7

### Direct product of C2 and Q8.D14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×Q8.D14
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×Q8⋊2D7 — C2×Q8.D14
 Lower central C7 — C14 — C28 — C2×Q8.D14
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C2×Q8.D14
G = < a,b,c,d,e | a2=b4=e2=1, c2=d14=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=b2d13 >

Subgroups: 1380 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C4○D4, Dic7, C28, C28, D14, D14, C2×C14, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, C7⋊C8, C56, C4×D7, C4×D7, D28, D28, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C22×D7, C2×C4○D8, C8×D7, D56, C2×C7⋊C8, Q8⋊D7, C2×C56, C7×Q16, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, Q82D7, Q82D7, Q8×C14, D7×C2×C8, C2×D56, Q8.D14, C2×Q8⋊D7, C14×Q16, C2×Q82D7, C2×Q8.D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C4○D8, C22×D4, C22×D7, C2×C4○D8, D4×D7, C23×D7, Q8.D14, C2×D4×D7, C2×Q8.D14

Smallest permutation representation of C2×Q8.D14
On 224 points
Generators in S224
(1 144)(2 145)(3 146)(4 147)(5 148)(6 149)(7 150)(8 151)(9 152)(10 153)(11 154)(12 155)(13 156)(14 157)(15 158)(16 159)(17 160)(18 161)(19 162)(20 163)(21 164)(22 165)(23 166)(24 167)(25 168)(26 141)(27 142)(28 143)(29 196)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 177)(39 178)(40 179)(41 180)(42 181)(43 182)(44 183)(45 184)(46 185)(47 186)(48 187)(49 188)(50 189)(51 190)(52 191)(53 192)(54 193)(55 194)(56 195)(57 137)(58 138)(59 139)(60 140)(61 113)(62 114)(63 115)(64 116)(65 117)(66 118)(67 119)(68 120)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 220)(86 221)(87 222)(88 223)(89 224)(90 197)(91 198)(92 199)(93 200)(94 201)(95 202)(96 203)(97 204)(98 205)(99 206)(100 207)(101 208)(102 209)(103 210)(104 211)(105 212)(106 213)(107 214)(108 215)(109 216)(110 217)(111 218)(112 219)
(1 95 15 109)(2 110 16 96)(3 97 17 111)(4 112 18 98)(5 99 19 85)(6 86 20 100)(7 101 21 87)(8 88 22 102)(9 103 23 89)(10 90 24 104)(11 105 25 91)(12 92 26 106)(13 107 27 93)(14 94 28 108)(29 126 43 140)(30 113 44 127)(31 128 45 114)(32 115 46 129)(33 130 47 116)(34 117 48 131)(35 132 49 118)(36 119 50 133)(37 134 51 120)(38 121 52 135)(39 136 53 122)(40 123 54 137)(41 138 55 124)(42 125 56 139)(57 179 71 193)(58 194 72 180)(59 181 73 195)(60 196 74 182)(61 183 75 169)(62 170 76 184)(63 185 77 171)(64 172 78 186)(65 187 79 173)(66 174 80 188)(67 189 81 175)(68 176 82 190)(69 191 83 177)(70 178 84 192)(141 213 155 199)(142 200 156 214)(143 215 157 201)(144 202 158 216)(145 217 159 203)(146 204 160 218)(147 219 161 205)(148 206 162 220)(149 221 163 207)(150 208 164 222)(151 223 165 209)(152 210 166 224)(153 197 167 211)(154 212 168 198)
(1 71 15 57)(2 194 16 180)(3 73 17 59)(4 196 18 182)(5 75 19 61)(6 170 20 184)(7 77 21 63)(8 172 22 186)(9 79 23 65)(10 174 24 188)(11 81 25 67)(12 176 26 190)(13 83 27 69)(14 178 28 192)(29 161 43 147)(30 206 44 220)(31 163 45 149)(32 208 46 222)(33 165 47 151)(34 210 48 224)(35 167 49 153)(36 212 50 198)(37 141 51 155)(38 214 52 200)(39 143 53 157)(40 216 54 202)(41 145 55 159)(42 218 56 204)(58 96 72 110)(60 98 74 112)(62 100 76 86)(64 102 78 88)(66 104 80 90)(68 106 82 92)(70 108 84 94)(85 169 99 183)(87 171 101 185)(89 173 103 187)(91 175 105 189)(93 177 107 191)(95 179 109 193)(97 181 111 195)(113 148 127 162)(114 207 128 221)(115 150 129 164)(116 209 130 223)(117 152 131 166)(118 211 132 197)(119 154 133 168)(120 213 134 199)(121 156 135 142)(122 215 136 201)(123 158 137 144)(124 217 138 203)(125 160 139 146)(126 219 140 205)
(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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(51 56)(52 55)(53 54)(57 84)(58 83)(59 82)(60 81)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(85 90)(86 89)(87 88)(91 112)(92 111)(93 110)(94 109)(95 108)(96 107)(97 106)(98 105)(99 104)(100 103)(101 102)(113 132)(114 131)(115 130)(116 129)(117 128)(118 127)(119 126)(120 125)(121 124)(122 123)(133 140)(134 139)(135 138)(136 137)(141 160)(142 159)(143 158)(144 157)(145 156)(146 155)(147 154)(148 153)(149 152)(150 151)(161 168)(162 167)(163 166)(164 165)(169 188)(170 187)(171 186)(172 185)(173 184)(174 183)(175 182)(176 181)(177 180)(178 179)(189 196)(190 195)(191 194)(192 193)(197 220)(198 219)(199 218)(200 217)(201 216)(202 215)(203 214)(204 213)(205 212)(206 211)(207 210)(208 209)(221 224)(222 223)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 28A ··· 28F 28G ··· 28R 56A ··· 56L order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 1 1 14 14 28 28 28 28 2 2 4 4 4 4 7 7 7 7 2 2 2 2 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4 8 ··· 8 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 C4○D8 D4×D7 D4×D7 Q8.D14 kernel C2×Q8.D14 D7×C2×C8 C2×D56 Q8.D14 C2×Q8⋊D7 C14×Q16 C2×Q8⋊2D7 C4×D7 C2×Dic7 C22×D7 C2×Q16 C2×C8 Q16 C2×Q8 C14 C4 C22 C2 # reps 1 1 1 8 2 1 2 2 1 1 3 3 12 6 8 3 3 12

Matrix representation of C2×Q8.D14 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 1 0 0 0 0 98 0 0 0 53 15
,
 1 0 0 0 0 1 0 0 0 0 88 69 0 0 81 25
,
 104 80 0 0 33 33 0 0 0 0 111 1 0 0 108 2
,
 104 80 0 0 112 9 0 0 0 0 2 112 0 0 3 111
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,53,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,88,81,0,0,69,25],[104,33,0,0,80,33,0,0,0,0,111,108,0,0,1,2],[104,112,0,0,80,9,0,0,0,0,2,3,0,0,112,111] >;

C2×Q8.D14 in GAP, Magma, Sage, TeX

C_2\times Q_8.D_{14}
% in TeX

G:=Group("C2xQ8.D14");
// GroupNames label

G:=SmallGroup(448,1218);
// by ID

G=gap.SmallGroup(448,1218);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,184,1123,185,136,438,235,102,18822]);
// Polycyclic

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

׿
×
𝔽