direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4.Dic7, C28.74C24, C24.5Dic7, C7⋊C8⋊13C23, (C23×C4).9D7, C14⋊3(C2×M4(2)), (C2×C14)⋊9M4(2), C4.73(C23×D7), C7⋊3(C22×M4(2)), C14.41(C23×C4), (C23×C14).11C4, (C22×C28).28C4, (C23×C28).16C2, C2.3(C23×Dic7), (C2×C28).799C23, C28.180(C22×C4), (C22×C4).472D14, C23.35(C2×Dic7), (C22×C4).18Dic7, C4.38(C22×Dic7), (C22×C28).546C22, C22.28(C22×Dic7), (C22×C7⋊C8)⋊23C2, (C2×C7⋊C8)⋊48C22, (C2×C28).300(C2×C4), (C2×C4).86(C2×Dic7), (C2×C4).827(C22×D7), (C22×C14).140(C2×C4), (C2×C14).205(C22×C4), SmallGroup(448,1234)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C4.Dic7
G = < a,b,c,d,e | a2=b2=c4=1, d14=c2, e2=c2d7, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d13 >
Subgroups: 644 in 298 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C22×C8, C2×M4(2), C23×C4, C7⋊C8, C2×C28, C22×C14, C22×C14, C22×C14, C22×M4(2), C2×C7⋊C8, C4.Dic7, C22×C28, C22×C28, C23×C14, C22×C7⋊C8, C2×C4.Dic7, C23×C28, C22×C4.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, M4(2), C22×C4, C24, Dic7, D14, C2×M4(2), C23×C4, C2×Dic7, C22×D7, C22×M4(2), C4.Dic7, C22×Dic7, C23×D7, C2×C4.Dic7, C23×Dic7, C22×C4.Dic7
►On 224 points
Generators in S224
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 57)(16 58)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(113 185)(114 186)(115 187)(116 188)(117 189)(118 190)(119 191)(120 192)(121 193)(122 194)(123 195)(124 196)(125 169)(126 170)(127 171)(128 172)(129 173)(130 174)(131 175)(132 176)(133 177)(134 178)(135 179)(136 180)(137 181)(138 182)(139 183)(140 184)(141 220)(142 221)(143 222)(144 223)(145 224)(146 197)(147 198)(148 199)(149 200)(150 201)(151 202)(152 203)(153 204)(154 205)(155 206)(156 207)(157 208)(158 209)(159 210)(160 211)(161 212)(162 213)(163 214)(164 215)(165 216)(166 217)(167 218)(168 219)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 99)(58 100)(59 101)(60 102)(61 103)(62 104)(63 105)(64 106)(65 107)(66 108)(67 109)(68 110)(69 111)(70 112)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 91)(78 92)(79 93)(80 94)(81 95)(82 96)(83 97)(84 98)(113 144)(114 145)(115 146)(116 147)(117 148)(118 149)(119 150)(120 151)(121 152)(122 153)(123 154)(124 155)(125 156)(126 157)(127 158)(128 159)(129 160)(130 161)(131 162)(132 163)(133 164)(134 165)(135 166)(136 167)(137 168)(138 141)(139 142)(140 143)(169 207)(170 208)(171 209)(172 210)(173 211)(174 212)(175 213)(176 214)(177 215)(178 216)(179 217)(180 218)(181 219)(182 220)(183 221)(184 222)(185 223)(186 224)(187 197)(188 198)(189 199)(190 200)(191 201)(192 202)(193 203)(194 204)(195 205)(196 206)
(1 43 15 29)(2 44 16 30)(3 45 17 31)(4 46 18 32)(5 47 19 33)(6 48 20 34)(7 49 21 35)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 92 71 106)(58 93 72 107)(59 94 73 108)(60 95 74 109)(61 96 75 110)(62 97 76 111)(63 98 77 112)(64 99 78 85)(65 100 79 86)(66 101 80 87)(67 102 81 88)(68 103 82 89)(69 104 83 90)(70 105 84 91)(113 151 127 165)(114 152 128 166)(115 153 129 167)(116 154 130 168)(117 155 131 141)(118 156 132 142)(119 157 133 143)(120 158 134 144)(121 159 135 145)(122 160 136 146)(123 161 137 147)(124 162 138 148)(125 163 139 149)(126 164 140 150)(169 214 183 200)(170 215 184 201)(171 216 185 202)(172 217 186 203)(173 218 187 204)(174 219 188 205)(175 220 189 206)(176 221 190 207)(177 222 191 208)(178 223 192 209)(179 224 193 210)(180 197 194 211)(181 198 195 212)(182 199 196 213)
(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)
(1 133 22 126 15 119 8 140)(2 118 23 139 16 132 9 125)(3 131 24 124 17 117 10 138)(4 116 25 137 18 130 11 123)(5 129 26 122 19 115 12 136)(6 114 27 135 20 128 13 121)(7 127 28 120 21 113 14 134)(29 143 50 164 43 157 36 150)(30 156 51 149 44 142 37 163)(31 141 52 162 45 155 38 148)(32 154 53 147 46 168 39 161)(33 167 54 160 47 153 40 146)(34 152 55 145 48 166 41 159)(35 165 56 158 49 151 42 144)(57 191 78 184 71 177 64 170)(58 176 79 169 72 190 65 183)(59 189 80 182 73 175 66 196)(60 174 81 195 74 188 67 181)(61 187 82 180 75 173 68 194)(62 172 83 193 76 186 69 179)(63 185 84 178 77 171 70 192)(85 215 106 208 99 201 92 222)(86 200 107 221 100 214 93 207)(87 213 108 206 101 199 94 220)(88 198 109 219 102 212 95 205)(89 211 110 204 103 197 96 218)(90 224 111 217 104 210 97 203)(91 209 112 202 105 223 98 216)
G:=sub<Sym(224)| (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)(141,220)(142,221)(143,222)(144,223)(145,224)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,141)(139,142)(140,143)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218)(181,219)(182,220)(183,221)(184,222)(185,223)(186,224)(187,197)(188,198)(189,199)(190,200)(191,201)(192,202)(193,203)(194,204)(195,205)(196,206), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,92,71,106)(58,93,72,107)(59,94,73,108)(60,95,74,109)(61,96,75,110)(62,97,76,111)(63,98,77,112)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91)(113,151,127,165)(114,152,128,166)(115,153,129,167)(116,154,130,168)(117,155,131,141)(118,156,132,142)(119,157,133,143)(120,158,134,144)(121,159,135,145)(122,160,136,146)(123,161,137,147)(124,162,138,148)(125,163,139,149)(126,164,140,150)(169,214,183,200)(170,215,184,201)(171,216,185,202)(172,217,186,203)(173,218,187,204)(174,219,188,205)(175,220,189,206)(176,221,190,207)(177,222,191,208)(178,223,192,209)(179,224,193,210)(180,197,194,211)(181,198,195,212)(182,199,196,213), (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), (1,133,22,126,15,119,8,140)(2,118,23,139,16,132,9,125)(3,131,24,124,17,117,10,138)(4,116,25,137,18,130,11,123)(5,129,26,122,19,115,12,136)(6,114,27,135,20,128,13,121)(7,127,28,120,21,113,14,134)(29,143,50,164,43,157,36,150)(30,156,51,149,44,142,37,163)(31,141,52,162,45,155,38,148)(32,154,53,147,46,168,39,161)(33,167,54,160,47,153,40,146)(34,152,55,145,48,166,41,159)(35,165,56,158,49,151,42,144)(57,191,78,184,71,177,64,170)(58,176,79,169,72,190,65,183)(59,189,80,182,73,175,66,196)(60,174,81,195,74,188,67,181)(61,187,82,180,75,173,68,194)(62,172,83,193,76,186,69,179)(63,185,84,178,77,171,70,192)(85,215,106,208,99,201,92,222)(86,200,107,221,100,214,93,207)(87,213,108,206,101,199,94,220)(88,198,109,219,102,212,95,205)(89,211,110,204,103,197,96,218)(90,224,111,217,104,210,97,203)(91,209,112,202,105,223,98,216)>;
G:=Group( (1,71)(2,72)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(113,185)(114,186)(115,187)(116,188)(117,189)(118,190)(119,191)(120,192)(121,193)(122,194)(123,195)(124,196)(125,169)(126,170)(127,171)(128,172)(129,173)(130,174)(131,175)(132,176)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)(141,220)(142,221)(143,222)(144,223)(145,224)(146,197)(147,198)(148,199)(149,200)(150,201)(151,202)(152,203)(153,204)(154,205)(155,206)(156,207)(157,208)(158,209)(159,210)(160,211)(161,212)(162,213)(163,214)(164,215)(165,216)(166,217)(167,218)(168,219), (1,50)(2,51)(3,52)(4,53)(5,54)(6,55)(7,56)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,36)(16,37)(17,38)(18,39)(19,40)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,99)(58,100)(59,101)(60,102)(61,103)(62,104)(63,105)(64,106)(65,107)(66,108)(67,109)(68,110)(69,111)(70,112)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,91)(78,92)(79,93)(80,94)(81,95)(82,96)(83,97)(84,98)(113,144)(114,145)(115,146)(116,147)(117,148)(118,149)(119,150)(120,151)(121,152)(122,153)(123,154)(124,155)(125,156)(126,157)(127,158)(128,159)(129,160)(130,161)(131,162)(132,163)(133,164)(134,165)(135,166)(136,167)(137,168)(138,141)(139,142)(140,143)(169,207)(170,208)(171,209)(172,210)(173,211)(174,212)(175,213)(176,214)(177,215)(178,216)(179,217)(180,218)(181,219)(182,220)(183,221)(184,222)(185,223)(186,224)(187,197)(188,198)(189,199)(190,200)(191,201)(192,202)(193,203)(194,204)(195,205)(196,206), (1,43,15,29)(2,44,16,30)(3,45,17,31)(4,46,18,32)(5,47,19,33)(6,48,20,34)(7,49,21,35)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,92,71,106)(58,93,72,107)(59,94,73,108)(60,95,74,109)(61,96,75,110)(62,97,76,111)(63,98,77,112)(64,99,78,85)(65,100,79,86)(66,101,80,87)(67,102,81,88)(68,103,82,89)(69,104,83,90)(70,105,84,91)(113,151,127,165)(114,152,128,166)(115,153,129,167)(116,154,130,168)(117,155,131,141)(118,156,132,142)(119,157,133,143)(120,158,134,144)(121,159,135,145)(122,160,136,146)(123,161,137,147)(124,162,138,148)(125,163,139,149)(126,164,140,150)(169,214,183,200)(170,215,184,201)(171,216,185,202)(172,217,186,203)(173,218,187,204)(174,219,188,205)(175,220,189,206)(176,221,190,207)(177,222,191,208)(178,223,192,209)(179,224,193,210)(180,197,194,211)(181,198,195,212)(182,199,196,213), (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), (1,133,22,126,15,119,8,140)(2,118,23,139,16,132,9,125)(3,131,24,124,17,117,10,138)(4,116,25,137,18,130,11,123)(5,129,26,122,19,115,12,136)(6,114,27,135,20,128,13,121)(7,127,28,120,21,113,14,134)(29,143,50,164,43,157,36,150)(30,156,51,149,44,142,37,163)(31,141,52,162,45,155,38,148)(32,154,53,147,46,168,39,161)(33,167,54,160,47,153,40,146)(34,152,55,145,48,166,41,159)(35,165,56,158,49,151,42,144)(57,191,78,184,71,177,64,170)(58,176,79,169,72,190,65,183)(59,189,80,182,73,175,66,196)(60,174,81,195,74,188,67,181)(61,187,82,180,75,173,68,194)(62,172,83,193,76,186,69,179)(63,185,84,178,77,171,70,192)(85,215,106,208,99,201,92,222)(86,200,107,221,100,214,93,207)(87,213,108,206,101,199,94,220)(88,198,109,219,102,212,95,205)(89,211,110,204,103,197,96,218)(90,224,111,217,104,210,97,203)(91,209,112,202,105,223,98,216) );
G=PermutationGroup([[(1,71),(2,72),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,57),(16,58),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,65),(24,66),(25,67),(26,68),(27,69),(28,70),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(113,185),(114,186),(115,187),(116,188),(117,189),(118,190),(119,191),(120,192),(121,193),(122,194),(123,195),(124,196),(125,169),(126,170),(127,171),(128,172),(129,173),(130,174),(131,175),(132,176),(133,177),(134,178),(135,179),(136,180),(137,181),(138,182),(139,183),(140,184),(141,220),(142,221),(143,222),(144,223),(145,224),(146,197),(147,198),(148,199),(149,200),(150,201),(151,202),(152,203),(153,204),(154,205),(155,206),(156,207),(157,208),(158,209),(159,210),(160,211),(161,212),(162,213),(163,214),(164,215),(165,216),(166,217),(167,218),(168,219)], [(1,50),(2,51),(3,52),(4,53),(5,54),(6,55),(7,56),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,36),(16,37),(17,38),(18,39),(19,40),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,99),(58,100),(59,101),(60,102),(61,103),(62,104),(63,105),(64,106),(65,107),(66,108),(67,109),(68,110),(69,111),(70,112),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,91),(78,92),(79,93),(80,94),(81,95),(82,96),(83,97),(84,98),(113,144),(114,145),(115,146),(116,147),(117,148),(118,149),(119,150),(120,151),(121,152),(122,153),(123,154),(124,155),(125,156),(126,157),(127,158),(128,159),(129,160),(130,161),(131,162),(132,163),(133,164),(134,165),(135,166),(136,167),(137,168),(138,141),(139,142),(140,143),(169,207),(170,208),(171,209),(172,210),(173,211),(174,212),(175,213),(176,214),(177,215),(178,216),(179,217),(180,218),(181,219),(182,220),(183,221),(184,222),(185,223),(186,224),(187,197),(188,198),(189,199),(190,200),(191,201),(192,202),(193,203),(194,204),(195,205),(196,206)], [(1,43,15,29),(2,44,16,30),(3,45,17,31),(4,46,18,32),(5,47,19,33),(6,48,20,34),(7,49,21,35),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,92,71,106),(58,93,72,107),(59,94,73,108),(60,95,74,109),(61,96,75,110),(62,97,76,111),(63,98,77,112),(64,99,78,85),(65,100,79,86),(66,101,80,87),(67,102,81,88),(68,103,82,89),(69,104,83,90),(70,105,84,91),(113,151,127,165),(114,152,128,166),(115,153,129,167),(116,154,130,168),(117,155,131,141),(118,156,132,142),(119,157,133,143),(120,158,134,144),(121,159,135,145),(122,160,136,146),(123,161,137,147),(124,162,138,148),(125,163,139,149),(126,164,140,150),(169,214,183,200),(170,215,184,201),(171,216,185,202),(172,217,186,203),(173,218,187,204),(174,219,188,205),(175,220,189,206),(176,221,190,207),(177,222,191,208),(178,223,192,209),(179,224,193,210),(180,197,194,211),(181,198,195,212),(182,199,196,213)], [(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)], [(1,133,22,126,15,119,8,140),(2,118,23,139,16,132,9,125),(3,131,24,124,17,117,10,138),(4,116,25,137,18,130,11,123),(5,129,26,122,19,115,12,136),(6,114,27,135,20,128,13,121),(7,127,28,120,21,113,14,134),(29,143,50,164,43,157,36,150),(30,156,51,149,44,142,37,163),(31,141,52,162,45,155,38,148),(32,154,53,147,46,168,39,161),(33,167,54,160,47,153,40,146),(34,152,55,145,48,166,41,159),(35,165,56,158,49,151,42,144),(57,191,78,184,71,177,64,170),(58,176,79,169,72,190,65,183),(59,189,80,182,73,175,66,196),(60,174,81,195,74,188,67,181),(61,187,82,180,75,173,68,194),(62,172,83,193,76,186,69,179),(63,185,84,178,77,171,70,192),(85,215,106,208,99,201,92,222),(86,200,107,221,100,214,93,207),(87,213,108,206,101,199,94,220),(88,198,109,219,102,212,95,205),(89,211,110,204,103,197,96,218),(90,224,111,217,104,210,97,203),(91,209,112,202,105,223,98,216)]])
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 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D7 | M4(2) | Dic7 | D14 | Dic7 | C4.Dic7 |
| kernel | C22×C4.Dic7 | C22×C7⋊C8 | C2×C4.Dic7 | C23×C28 | C22×C28 | C23×C14 | C23×C4 | C2×C14 | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 12 | 1 | 14 | 2 | 3 | 8 | 21 | 21 | 3 | 48 |
Matrix representation of C22×C4.Dic7 ►in GL5(𝔽113)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 7 | 98 | 0 | 0 |
| 0 | 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 37 | 15 |
| 112 | 0 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 0 | 111 | 0 |
| 0 | 0 | 0 | 69 | 57 |
| 98 | 0 | 0 | 0 | 0 |
| 0 | 62 | 41 | 0 | 0 |
| 0 | 83 | 51 | 0 | 0 |
| 0 | 0 | 0 | 14 | 48 |
| 0 | 0 | 0 | 8 | 99 |
G:=sub<GL(5,GF(113))| [1,0,0,0,0,0,112,0,0,0,0,0,112,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[112,0,0,0,0,0,15,7,0,0,0,0,98,0,0,0,0,0,98,37,0,0,0,0,15],[112,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,111,69,0,0,0,0,57],[98,0,0,0,0,0,62,83,0,0,0,41,51,0,0,0,0,0,14,8,0,0,0,48,99] >;
C22×C4.Dic7 in GAP, Magma, Sage, TeX
C_2^2\times C_4.{\rm Dic}_7 % in TeX
G:=Group("C2^2xC4.Dic7"); // GroupNames label
G:=SmallGroup(448,1234);
// by ID
G=gap.SmallGroup(448,1234);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,1123,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^14=c^2,e^2=c^2*d^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^13>;
// generators/relations