direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D14⋊D4, C24.26D14, D14⋊1(C2×D4), Dic7⋊6(C2×D4), (C22×D7)⋊9D4, C14⋊1(C4⋊D4), C22⋊C4⋊39D14, (C2×Dic7)⋊19D4, (C22×D28)⋊6C2, D14⋊C4⋊58C22, (C2×D28)⋊44C22, (C2×C14).33C24, C22.127(D4×D7), C14.36(C22×D4), (C2×C28).573C23, Dic7⋊C4⋊48C22, (C22×C4).170D14, (C22×D7).6C23, C23.80(C22×D7), C22.72(C23×D7), C22.73(C4○D28), (C23×C14).59C22, (C23×D7).96C22, (C22×C28).353C22, (C22×C14).125C23, (C2×Dic7).179C23, (C22×Dic7).77C22, C7⋊1(C2×C4⋊D4), C2.10(C2×D4×D7), (C2×C4×D7)⋊66C22, (D7×C22×C4)⋊18C2, (C2×D14⋊C4)⋊31C2, (C2×C22⋊C4)⋊12D7, C14.13(C2×C4○D4), C2.15(C2×C4○D28), (C22×C7⋊D4)⋊4C2, (C14×C22⋊C4)⋊17C2, (C2×Dic7⋊C4)⋊22C2, (C2×C14).382(C2×D4), (C2×C7⋊D4)⋊35C22, (C7×C22⋊C4)⋊52C22, (C2×C4).134(C22×D7), (C2×C14).102(C4○D4), SmallGroup(448,942)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D14⋊D4
G = < a,b,c,d,e | a2=b14=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe=b-1, dcd-1=b12c, ece=b5c, ede=d-1 >
Subgroups: 2260 in 426 conjugacy classes, 127 normal (31 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C4⋊D4, Dic7⋊C4, D14⋊C4, C7×C22⋊C4, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, D14⋊D4, C2×Dic7⋊C4, C2×D14⋊C4, C14×C22⋊C4, D7×C22×C4, C22×D28, C22×C7⋊D4, C2×D14⋊D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C4⋊D4, C22×D4, C2×C4○D4, C22×D7, C2×C4⋊D4, C4○D28, D4×D7, C23×D7, D14⋊D4, C2×C4○D28, C2×D4×D7, C2×D14⋊D4
►On 224 points
Generators in S224
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 97)(14 98)(15 134)(16 135)(17 136)(18 137)(19 138)(20 139)(21 140)(22 127)(23 128)(24 129)(25 130)(26 131)(27 132)(28 133)(29 45)(30 46)(31 47)(32 48)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 43)(42 44)(57 146)(58 147)(59 148)(60 149)(61 150)(62 151)(63 152)(64 153)(65 154)(66 141)(67 142)(68 143)(69 144)(70 145)(71 103)(72 104)(73 105)(74 106)(75 107)(76 108)(77 109)(78 110)(79 111)(80 112)(81 99)(82 100)(83 101)(84 102)(113 194)(114 195)(115 196)(116 183)(117 184)(118 185)(119 186)(120 187)(121 188)(122 189)(123 190)(124 191)(125 192)(126 193)(155 174)(156 175)(157 176)(158 177)(159 178)(160 179)(161 180)(162 181)(163 182)(164 169)(165 170)(166 171)(167 172)(168 173)(197 218)(198 219)(199 220)(200 221)(201 222)(202 223)(203 224)(204 211)(205 212)(206 213)(207 214)(208 215)(209 216)(210 217)
(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 20)(16 19)(17 18)(21 28)(22 27)(23 26)(24 25)(29 39)(30 38)(31 37)(32 36)(33 35)(40 42)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(69 70)(71 72)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(99 108)(100 107)(101 106)(102 105)(103 104)(109 112)(110 111)(113 115)(116 126)(117 125)(118 124)(119 123)(120 122)(127 132)(128 131)(129 130)(133 140)(134 139)(135 138)(136 137)(141 148)(142 147)(143 146)(144 145)(149 154)(150 153)(151 152)(155 159)(156 158)(160 168)(161 167)(162 166)(163 165)(170 182)(171 181)(172 180)(173 179)(174 178)(175 177)(183 193)(184 192)(185 191)(186 190)(187 189)(194 196)(198 210)(199 209)(200 208)(201 207)(202 206)(203 205)(212 224)(213 223)(214 222)(215 221)(216 220)(217 219)
(1 70 18 111)(2 69 19 110)(3 68 20 109)(4 67 21 108)(5 66 22 107)(6 65 23 106)(7 64 24 105)(8 63 25 104)(9 62 26 103)(10 61 27 102)(11 60 28 101)(12 59 15 100)(13 58 16 99)(14 57 17 112)(29 113 213 163)(30 126 214 162)(31 125 215 161)(32 124 216 160)(33 123 217 159)(34 122 218 158)(35 121 219 157)(36 120 220 156)(37 119 221 155)(38 118 222 168)(39 117 223 167)(40 116 224 166)(41 115 211 165)(42 114 212 164)(43 196 204 170)(44 195 205 169)(45 194 206 182)(46 193 207 181)(47 192 208 180)(48 191 209 179)(49 190 210 178)(50 189 197 177)(51 188 198 176)(52 187 199 175)(53 186 200 174)(54 185 201 173)(55 184 202 172)(56 183 203 171)(71 93 151 131)(72 92 152 130)(73 91 153 129)(74 90 154 128)(75 89 141 127)(76 88 142 140)(77 87 143 139)(78 86 144 138)(79 85 145 137)(80 98 146 136)(81 97 147 135)(82 96 148 134)(83 95 149 133)(84 94 150 132)
(1 185)(2 184)(3 183)(4 196)(5 195)(6 194)(7 193)(8 192)(9 191)(10 190)(11 189)(12 188)(13 187)(14 186)(15 176)(16 175)(17 174)(18 173)(19 172)(20 171)(21 170)(22 169)(23 182)(24 181)(25 180)(26 179)(27 178)(28 177)(29 154)(30 153)(31 152)(32 151)(33 150)(34 149)(35 148)(36 147)(37 146)(38 145)(39 144)(40 143)(41 142)(42 141)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 60)(51 59)(52 58)(53 57)(54 70)(55 69)(56 68)(71 216)(72 215)(73 214)(74 213)(75 212)(76 211)(77 224)(78 223)(79 222)(80 221)(81 220)(82 219)(83 218)(84 217)(85 118)(86 117)(87 116)(88 115)(89 114)(90 113)(91 126)(92 125)(93 124)(94 123)(95 122)(96 121)(97 120)(98 119)(99 199)(100 198)(101 197)(102 210)(103 209)(104 208)(105 207)(106 206)(107 205)(108 204)(109 203)(110 202)(111 201)(112 200)(127 164)(128 163)(129 162)(130 161)(131 160)(132 159)(133 158)(134 157)(135 156)(136 155)(137 168)(138 167)(139 166)(140 165)
G:=sub<Sym(224)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,43)(42,44)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,141)(67,142)(68,143)(69,144)(70,145)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102)(113,194)(114,195)(115,196)(116,183)(117,184)(118,185)(119,186)(120,187)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(155,174)(156,175)(157,176)(158,177)(159,178)(160,179)(161,180)(162,181)(163,182)(164,169)(165,170)(166,171)(167,172)(168,173)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (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)(15,20)(16,19)(17,18)(21,28)(22,27)(23,26)(24,25)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111)(113,115)(116,126)(117,125)(118,124)(119,123)(120,122)(127,132)(128,131)(129,130)(133,140)(134,139)(135,138)(136,137)(141,148)(142,147)(143,146)(144,145)(149,154)(150,153)(151,152)(155,159)(156,158)(160,168)(161,167)(162,166)(163,165)(170,182)(171,181)(172,180)(173,179)(174,178)(175,177)(183,193)(184,192)(185,191)(186,190)(187,189)(194,196)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,70,18,111)(2,69,19,110)(3,68,20,109)(4,67,21,108)(5,66,22,107)(6,65,23,106)(7,64,24,105)(8,63,25,104)(9,62,26,103)(10,61,27,102)(11,60,28,101)(12,59,15,100)(13,58,16,99)(14,57,17,112)(29,113,213,163)(30,126,214,162)(31,125,215,161)(32,124,216,160)(33,123,217,159)(34,122,218,158)(35,121,219,157)(36,120,220,156)(37,119,221,155)(38,118,222,168)(39,117,223,167)(40,116,224,166)(41,115,211,165)(42,114,212,164)(43,196,204,170)(44,195,205,169)(45,194,206,182)(46,193,207,181)(47,192,208,180)(48,191,209,179)(49,190,210,178)(50,189,197,177)(51,188,198,176)(52,187,199,175)(53,186,200,174)(54,185,201,173)(55,184,202,172)(56,183,203,171)(71,93,151,131)(72,92,152,130)(73,91,153,129)(74,90,154,128)(75,89,141,127)(76,88,142,140)(77,87,143,139)(78,86,144,138)(79,85,145,137)(80,98,146,136)(81,97,147,135)(82,96,148,134)(83,95,149,133)(84,94,150,132), (1,185)(2,184)(3,183)(4,196)(5,195)(6,194)(7,193)(8,192)(9,191)(10,190)(11,189)(12,188)(13,187)(14,186)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,182)(24,181)(25,180)(26,179)(27,178)(28,177)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,70)(55,69)(56,68)(71,216)(72,215)(73,214)(74,213)(75,212)(76,211)(77,224)(78,223)(79,222)(80,221)(81,220)(82,219)(83,218)(84,217)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,199)(100,198)(101,197)(102,210)(103,209)(104,208)(105,207)(106,206)(107,205)(108,204)(109,203)(110,202)(111,201)(112,200)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,168)(138,167)(139,166)(140,165)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,97)(14,98)(15,134)(16,135)(17,136)(18,137)(19,138)(20,139)(21,140)(22,127)(23,128)(24,129)(25,130)(26,131)(27,132)(28,133)(29,45)(30,46)(31,47)(32,48)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(41,43)(42,44)(57,146)(58,147)(59,148)(60,149)(61,150)(62,151)(63,152)(64,153)(65,154)(66,141)(67,142)(68,143)(69,144)(70,145)(71,103)(72,104)(73,105)(74,106)(75,107)(76,108)(77,109)(78,110)(79,111)(80,112)(81,99)(82,100)(83,101)(84,102)(113,194)(114,195)(115,196)(116,183)(117,184)(118,185)(119,186)(120,187)(121,188)(122,189)(123,190)(124,191)(125,192)(126,193)(155,174)(156,175)(157,176)(158,177)(159,178)(160,179)(161,180)(162,181)(163,182)(164,169)(165,170)(166,171)(167,172)(168,173)(197,218)(198,219)(199,220)(200,221)(201,222)(202,223)(203,224)(204,211)(205,212)(206,213)(207,214)(208,215)(209,216)(210,217), (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)(15,20)(16,19)(17,18)(21,28)(22,27)(23,26)(24,25)(29,39)(30,38)(31,37)(32,36)(33,35)(40,42)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)(69,70)(71,72)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(99,108)(100,107)(101,106)(102,105)(103,104)(109,112)(110,111)(113,115)(116,126)(117,125)(118,124)(119,123)(120,122)(127,132)(128,131)(129,130)(133,140)(134,139)(135,138)(136,137)(141,148)(142,147)(143,146)(144,145)(149,154)(150,153)(151,152)(155,159)(156,158)(160,168)(161,167)(162,166)(163,165)(170,182)(171,181)(172,180)(173,179)(174,178)(175,177)(183,193)(184,192)(185,191)(186,190)(187,189)(194,196)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(212,224)(213,223)(214,222)(215,221)(216,220)(217,219), (1,70,18,111)(2,69,19,110)(3,68,20,109)(4,67,21,108)(5,66,22,107)(6,65,23,106)(7,64,24,105)(8,63,25,104)(9,62,26,103)(10,61,27,102)(11,60,28,101)(12,59,15,100)(13,58,16,99)(14,57,17,112)(29,113,213,163)(30,126,214,162)(31,125,215,161)(32,124,216,160)(33,123,217,159)(34,122,218,158)(35,121,219,157)(36,120,220,156)(37,119,221,155)(38,118,222,168)(39,117,223,167)(40,116,224,166)(41,115,211,165)(42,114,212,164)(43,196,204,170)(44,195,205,169)(45,194,206,182)(46,193,207,181)(47,192,208,180)(48,191,209,179)(49,190,210,178)(50,189,197,177)(51,188,198,176)(52,187,199,175)(53,186,200,174)(54,185,201,173)(55,184,202,172)(56,183,203,171)(71,93,151,131)(72,92,152,130)(73,91,153,129)(74,90,154,128)(75,89,141,127)(76,88,142,140)(77,87,143,139)(78,86,144,138)(79,85,145,137)(80,98,146,136)(81,97,147,135)(82,96,148,134)(83,95,149,133)(84,94,150,132), (1,185)(2,184)(3,183)(4,196)(5,195)(6,194)(7,193)(8,192)(9,191)(10,190)(11,189)(12,188)(13,187)(14,186)(15,176)(16,175)(17,174)(18,173)(19,172)(20,171)(21,170)(22,169)(23,182)(24,181)(25,180)(26,179)(27,178)(28,177)(29,154)(30,153)(31,152)(32,151)(33,150)(34,149)(35,148)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,60)(51,59)(52,58)(53,57)(54,70)(55,69)(56,68)(71,216)(72,215)(73,214)(74,213)(75,212)(76,211)(77,224)(78,223)(79,222)(80,221)(81,220)(82,219)(83,218)(84,217)(85,118)(86,117)(87,116)(88,115)(89,114)(90,113)(91,126)(92,125)(93,124)(94,123)(95,122)(96,121)(97,120)(98,119)(99,199)(100,198)(101,197)(102,210)(103,209)(104,208)(105,207)(106,206)(107,205)(108,204)(109,203)(110,202)(111,201)(112,200)(127,164)(128,163)(129,162)(130,161)(131,160)(132,159)(133,158)(134,157)(135,156)(136,155)(137,168)(138,167)(139,166)(140,165) );
G=PermutationGroup([[(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,97),(14,98),(15,134),(16,135),(17,136),(18,137),(19,138),(20,139),(21,140),(22,127),(23,128),(24,129),(25,130),(26,131),(27,132),(28,133),(29,45),(30,46),(31,47),(32,48),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(41,43),(42,44),(57,146),(58,147),(59,148),(60,149),(61,150),(62,151),(63,152),(64,153),(65,154),(66,141),(67,142),(68,143),(69,144),(70,145),(71,103),(72,104),(73,105),(74,106),(75,107),(76,108),(77,109),(78,110),(79,111),(80,112),(81,99),(82,100),(83,101),(84,102),(113,194),(114,195),(115,196),(116,183),(117,184),(118,185),(119,186),(120,187),(121,188),(122,189),(123,190),(124,191),(125,192),(126,193),(155,174),(156,175),(157,176),(158,177),(159,178),(160,179),(161,180),(162,181),(163,182),(164,169),(165,170),(166,171),(167,172),(168,173),(197,218),(198,219),(199,220),(200,221),(201,222),(202,223),(203,224),(204,211),(205,212),(206,213),(207,214),(208,215),(209,216),(210,217)], [(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),(15,20),(16,19),(17,18),(21,28),(22,27),(23,26),(24,25),(29,39),(30,38),(31,37),(32,36),(33,35),(40,42),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(57,68),(58,67),(59,66),(60,65),(61,64),(62,63),(69,70),(71,72),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(99,108),(100,107),(101,106),(102,105),(103,104),(109,112),(110,111),(113,115),(116,126),(117,125),(118,124),(119,123),(120,122),(127,132),(128,131),(129,130),(133,140),(134,139),(135,138),(136,137),(141,148),(142,147),(143,146),(144,145),(149,154),(150,153),(151,152),(155,159),(156,158),(160,168),(161,167),(162,166),(163,165),(170,182),(171,181),(172,180),(173,179),(174,178),(175,177),(183,193),(184,192),(185,191),(186,190),(187,189),(194,196),(198,210),(199,209),(200,208),(201,207),(202,206),(203,205),(212,224),(213,223),(214,222),(215,221),(216,220),(217,219)], [(1,70,18,111),(2,69,19,110),(3,68,20,109),(4,67,21,108),(5,66,22,107),(6,65,23,106),(7,64,24,105),(8,63,25,104),(9,62,26,103),(10,61,27,102),(11,60,28,101),(12,59,15,100),(13,58,16,99),(14,57,17,112),(29,113,213,163),(30,126,214,162),(31,125,215,161),(32,124,216,160),(33,123,217,159),(34,122,218,158),(35,121,219,157),(36,120,220,156),(37,119,221,155),(38,118,222,168),(39,117,223,167),(40,116,224,166),(41,115,211,165),(42,114,212,164),(43,196,204,170),(44,195,205,169),(45,194,206,182),(46,193,207,181),(47,192,208,180),(48,191,209,179),(49,190,210,178),(50,189,197,177),(51,188,198,176),(52,187,199,175),(53,186,200,174),(54,185,201,173),(55,184,202,172),(56,183,203,171),(71,93,151,131),(72,92,152,130),(73,91,153,129),(74,90,154,128),(75,89,141,127),(76,88,142,140),(77,87,143,139),(78,86,144,138),(79,85,145,137),(80,98,146,136),(81,97,147,135),(82,96,148,134),(83,95,149,133),(84,94,150,132)], [(1,185),(2,184),(3,183),(4,196),(5,195),(6,194),(7,193),(8,192),(9,191),(10,190),(11,189),(12,188),(13,187),(14,186),(15,176),(16,175),(17,174),(18,173),(19,172),(20,171),(21,170),(22,169),(23,182),(24,181),(25,180),(26,179),(27,178),(28,177),(29,154),(30,153),(31,152),(32,151),(33,150),(34,149),(35,148),(36,147),(37,146),(38,145),(39,144),(40,143),(41,142),(42,141),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,60),(51,59),(52,58),(53,57),(54,70),(55,69),(56,68),(71,216),(72,215),(73,214),(74,213),(75,212),(76,211),(77,224),(78,223),(79,222),(80,221),(81,220),(82,219),(83,218),(84,217),(85,118),(86,117),(87,116),(88,115),(89,114),(90,113),(91,126),(92,125),(93,124),(94,123),(95,122),(96,121),(97,120),(98,119),(99,199),(100,198),(101,197),(102,210),(103,209),(104,208),(105,207),(106,206),(107,205),(108,204),(109,203),(110,202),(111,201),(112,200),(127,164),(128,163),(129,162),(130,161),(131,160),(132,159),(133,158),(134,157),(135,156),(136,155),(137,168),(138,167),(139,166),(140,165)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D28 | D4×D7 |
| kernel | C2×D14⋊D4 | D14⋊D4 | C2×Dic7⋊C4 | C2×D14⋊C4 | C14×C22⋊C4 | D7×C22×C4 | C22×D28 | C22×C7⋊D4 | C2×Dic7 | C22×D7 | C2×C22⋊C4 | C2×C14 | C22⋊C4 | C22×C4 | C24 | C22 | C22 |
| # reps | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 3 | 4 | 12 | 6 | 3 | 24 | 12 |
Matrix representation of C2×D14⋊D4 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 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 |
| 25 | 4 | 0 | 0 | 0 | 0 |
| 25 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 25 | 0 | 0 |
| 0 | 0 | 4 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
| 25 | 11 | 0 | 0 | 0 | 0 |
| 25 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 13 | 0 | 0 |
| 0 | 0 | 19 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 4 | 18 | 0 | 0 | 0 | 0 |
| 4 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 22 | 0 | 0 |
| 0 | 0 | 26 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 1 |
| 0 | 0 | 0 | 0 | 7 | 20 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[25,25,0,0,0,0,4,11,0,0,0,0,0,0,4,4,0,0,0,0,25,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,18,28,0,0,0,0,0,0,28,0,0,0,0,0,11,1,0,0,0,0,0,0,28,18,0,0,0,0,0,1],[25,25,0,0,0,0,11,4,0,0,0,0,0,0,19,19,0,0,0,0,13,10,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,4,0,0,0,0,18,25,0,0,0,0,0,0,3,26,0,0,0,0,22,26,0,0,0,0,0,0,9,7,0,0,0,0,1,20] >;
C2×D14⋊D4 in GAP, Magma, Sage, TeX
C_2\times D_{14}\rtimes D_4 % in TeX
G:=Group("C2xD14:D4"); // GroupNames label
G:=SmallGroup(448,942);
// by ID
G=gap.SmallGroup(448,942);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,1571,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^14=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^12*c,e*c*e=b^5*c,e*d*e=d^-1>;
// generators/relations