direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C2×C14, C24⋊5C28, C25.3C14, (C23×C28)⋊5C2, (C23×C14)⋊6C4, (C23×C4)⋊4C14, C23⋊6(C2×C28), (C2×C28)⋊13C23, C2.1(C23×C28), (C24×C14).2C2, C23.58(C7×D4), C14.53(C23×C4), C24.31(C2×C14), C22⋊2(C22×C28), C22.57(D4×C14), (C2×C14).332C24, (C22×C28)⋊57C22, (C22×C14).219D4, C14.177(C22×D4), C22.5(C23×C14), C23.65(C22×C14), (C23×C14).88C22, (C22×C14).251C23, C2.1(D4×C2×C14), (C2×C14)⋊8(C22×C4), (C2×C4)⋊3(C22×C14), (C22×C4)⋊15(C2×C14), (C22×C14)⋊16(C2×C4), (C2×C14).679(C2×D4), SmallGroup(448,1295)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22⋊C4×C2×C14
G = < a,b,c,d,e | a2=b14=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 1010 in 674 conjugacy classes, 338 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C23×C4, C25, C2×C28, C2×C28, C22×C14, C22×C14, C22×C22⋊C4, C7×C22⋊C4, C22×C28, C22×C28, C23×C14, C23×C14, C23×C14, C14×C22⋊C4, C23×C28, C24×C14, C22⋊C4×C2×C14
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C24, C28, C2×C14, C2×C22⋊C4, C23×C4, C22×D4, C2×C28, C7×D4, C22×C14, C22×C22⋊C4, C7×C22⋊C4, C22×C28, D4×C14, C23×C14, C14×C22⋊C4, C23×C28, D4×C2×C14, C22⋊C4×C2×C14
►On 224 points
Generators in S224
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 71)(42 72)(43 134)(44 135)(45 136)(46 137)(47 138)(48 139)(49 140)(50 127)(51 128)(52 129)(53 130)(54 131)(55 132)(56 133)(57 143)(58 144)(59 145)(60 146)(61 147)(62 148)(63 149)(64 150)(65 151)(66 152)(67 153)(68 154)(69 141)(70 142)(85 204)(86 205)(87 206)(88 207)(89 208)(90 209)(91 210)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 185)(100 186)(101 187)(102 188)(103 189)(104 190)(105 191)(106 192)(107 193)(108 194)(109 195)(110 196)(111 183)(112 184)(113 160)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 155)(123 156)(124 157)(125 158)(126 159)(169 222)(170 223)(171 224)(172 211)(173 212)(174 213)(175 214)(176 215)(177 216)(178 217)(179 218)(180 219)(181 220)(182 221)
(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 86)(2 87)(3 88)(4 89)(5 90)(6 91)(7 92)(8 93)(9 94)(10 95)(11 96)(12 97)(13 98)(14 85)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 197)(22 198)(23 199)(24 200)(25 201)(26 202)(27 203)(28 204)(29 80)(30 81)(31 82)(32 83)(33 84)(34 71)(35 72)(36 73)(37 74)(38 75)(39 76)(40 77)(41 78)(42 79)(43 145)(44 146)(45 147)(46 148)(47 149)(48 150)(49 151)(50 152)(51 153)(52 154)(53 141)(54 142)(55 143)(56 144)(57 132)(58 133)(59 134)(60 135)(61 136)(62 137)(63 138)(64 139)(65 140)(66 127)(67 128)(68 129)(69 130)(70 131)(99 192)(100 193)(101 194)(102 195)(103 196)(104 183)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 191)(113 167)(114 168)(115 155)(116 156)(117 157)(118 158)(119 159)(120 160)(121 161)(122 162)(123 163)(124 164)(125 165)(126 166)(169 215)(170 216)(171 217)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 211)(180 212)(181 213)(182 214)
(1 198)(2 199)(3 200)(4 201)(5 202)(6 203)(7 204)(8 205)(9 206)(10 207)(11 208)(12 209)(13 210)(14 197)(15 93)(16 94)(17 95)(18 96)(19 97)(20 98)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 169)(30 170)(31 171)(32 172)(33 173)(34 174)(35 175)(36 176)(37 177)(38 178)(39 179)(40 180)(41 181)(42 182)(43 66)(44 67)(45 68)(46 69)(47 70)(48 57)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 65)(71 220)(72 221)(73 222)(74 223)(75 224)(76 211)(77 212)(78 213)(79 214)(80 215)(81 216)(82 217)(83 218)(84 219)(99 124)(100 125)(101 126)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)(109 120)(110 121)(111 122)(112 123)(127 145)(128 146)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 153)(136 154)(137 141)(138 142)(139 143)(140 144)(155 183)(156 184)(157 185)(158 186)(159 187)(160 188)(161 189)(162 190)(163 191)(164 192)(165 193)(166 194)(167 195)(168 196)
(1 211 146 189)(2 212 147 190)(3 213 148 191)(4 214 149 192)(5 215 150 193)(6 216 151 194)(7 217 152 195)(8 218 153 196)(9 219 154 183)(10 220 141 184)(11 221 142 185)(12 222 143 186)(13 223 144 187)(14 224 145 188)(15 172 60 103)(16 173 61 104)(17 174 62 105)(18 175 63 106)(19 176 64 107)(20 177 65 108)(21 178 66 109)(22 179 67 110)(23 180 68 111)(24 181 69 112)(25 182 70 99)(26 169 57 100)(27 170 58 101)(28 171 59 102)(29 48 125 90)(30 49 126 91)(31 50 113 92)(32 51 114 93)(33 52 115 94)(34 53 116 95)(35 54 117 96)(36 55 118 97)(37 56 119 98)(38 43 120 85)(39 44 121 86)(40 45 122 87)(41 46 123 88)(42 47 124 89)(71 137 156 207)(72 138 157 208)(73 139 158 209)(74 140 159 210)(75 127 160 197)(76 128 161 198)(77 129 162 199)(78 130 163 200)(79 131 164 201)(80 132 165 202)(81 133 166 203)(82 134 167 204)(83 135 168 205)(84 136 155 206)
G:=sub<Sym(224)| (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,71)(42,72)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,141)(70,142)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,183)(112,184)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,155)(123,156)(124,157)(125,158)(126,159)(169,222)(170,223)(171,224)(172,211)(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)(182,221), (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), (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,85)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,197)(22,198)(23,199)(24,200)(25,201)(26,202)(27,203)(28,204)(29,80)(30,81)(31,82)(32,83)(33,84)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,141)(54,142)(55,143)(56,144)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,127)(67,128)(68,129)(69,130)(70,131)(99,192)(100,193)(101,194)(102,195)(103,196)(104,183)(105,184)(106,185)(107,186)(108,187)(109,188)(110,189)(111,190)(112,191)(113,167)(114,168)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,211)(180,212)(181,213)(182,214), (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,197)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,66)(44,67)(45,68)(46,69)(47,70)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(71,220)(72,221)(73,222)(74,223)(75,224)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(99,124)(100,125)(101,126)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)(111,122)(112,123)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,141)(138,142)(139,143)(140,144)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196), (1,211,146,189)(2,212,147,190)(3,213,148,191)(4,214,149,192)(5,215,150,193)(6,216,151,194)(7,217,152,195)(8,218,153,196)(9,219,154,183)(10,220,141,184)(11,221,142,185)(12,222,143,186)(13,223,144,187)(14,224,145,188)(15,172,60,103)(16,173,61,104)(17,174,62,105)(18,175,63,106)(19,176,64,107)(20,177,65,108)(21,178,66,109)(22,179,67,110)(23,180,68,111)(24,181,69,112)(25,182,70,99)(26,169,57,100)(27,170,58,101)(28,171,59,102)(29,48,125,90)(30,49,126,91)(31,50,113,92)(32,51,114,93)(33,52,115,94)(34,53,116,95)(35,54,117,96)(36,55,118,97)(37,56,119,98)(38,43,120,85)(39,44,121,86)(40,45,122,87)(41,46,123,88)(42,47,124,89)(71,137,156,207)(72,138,157,208)(73,139,158,209)(74,140,159,210)(75,127,160,197)(76,128,161,198)(77,129,162,199)(78,130,163,200)(79,131,164,201)(80,132,165,202)(81,133,166,203)(82,134,167,204)(83,135,168,205)(84,136,155,206)>;
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,73)(30,74)(31,75)(32,76)(33,77)(34,78)(35,79)(36,80)(37,81)(38,82)(39,83)(40,84)(41,71)(42,72)(43,134)(44,135)(45,136)(46,137)(47,138)(48,139)(49,140)(50,127)(51,128)(52,129)(53,130)(54,131)(55,132)(56,133)(57,143)(58,144)(59,145)(60,146)(61,147)(62,148)(63,149)(64,150)(65,151)(66,152)(67,153)(68,154)(69,141)(70,142)(85,204)(86,205)(87,206)(88,207)(89,208)(90,209)(91,210)(92,197)(93,198)(94,199)(95,200)(96,201)(97,202)(98,203)(99,185)(100,186)(101,187)(102,188)(103,189)(104,190)(105,191)(106,192)(107,193)(108,194)(109,195)(110,196)(111,183)(112,184)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,155)(123,156)(124,157)(125,158)(126,159)(169,222)(170,223)(171,224)(172,211)(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)(182,221), (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), (1,86)(2,87)(3,88)(4,89)(5,90)(6,91)(7,92)(8,93)(9,94)(10,95)(11,96)(12,97)(13,98)(14,85)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,197)(22,198)(23,199)(24,200)(25,201)(26,202)(27,203)(28,204)(29,80)(30,81)(31,82)(32,83)(33,84)(34,71)(35,72)(36,73)(37,74)(38,75)(39,76)(40,77)(41,78)(42,79)(43,145)(44,146)(45,147)(46,148)(47,149)(48,150)(49,151)(50,152)(51,153)(52,154)(53,141)(54,142)(55,143)(56,144)(57,132)(58,133)(59,134)(60,135)(61,136)(62,137)(63,138)(64,139)(65,140)(66,127)(67,128)(68,129)(69,130)(70,131)(99,192)(100,193)(101,194)(102,195)(103,196)(104,183)(105,184)(106,185)(107,186)(108,187)(109,188)(110,189)(111,190)(112,191)(113,167)(114,168)(115,155)(116,156)(117,157)(118,158)(119,159)(120,160)(121,161)(122,162)(123,163)(124,164)(125,165)(126,166)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,211)(180,212)(181,213)(182,214), (1,198)(2,199)(3,200)(4,201)(5,202)(6,203)(7,204)(8,205)(9,206)(10,207)(11,208)(12,209)(13,210)(14,197)(15,93)(16,94)(17,95)(18,96)(19,97)(20,98)(21,85)(22,86)(23,87)(24,88)(25,89)(26,90)(27,91)(28,92)(29,169)(30,170)(31,171)(32,172)(33,173)(34,174)(35,175)(36,176)(37,177)(38,178)(39,179)(40,180)(41,181)(42,182)(43,66)(44,67)(45,68)(46,69)(47,70)(48,57)(49,58)(50,59)(51,60)(52,61)(53,62)(54,63)(55,64)(56,65)(71,220)(72,221)(73,222)(74,223)(75,224)(76,211)(77,212)(78,213)(79,214)(80,215)(81,216)(82,217)(83,218)(84,219)(99,124)(100,125)(101,126)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)(111,122)(112,123)(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(136,154)(137,141)(138,142)(139,143)(140,144)(155,183)(156,184)(157,185)(158,186)(159,187)(160,188)(161,189)(162,190)(163,191)(164,192)(165,193)(166,194)(167,195)(168,196), (1,211,146,189)(2,212,147,190)(3,213,148,191)(4,214,149,192)(5,215,150,193)(6,216,151,194)(7,217,152,195)(8,218,153,196)(9,219,154,183)(10,220,141,184)(11,221,142,185)(12,222,143,186)(13,223,144,187)(14,224,145,188)(15,172,60,103)(16,173,61,104)(17,174,62,105)(18,175,63,106)(19,176,64,107)(20,177,65,108)(21,178,66,109)(22,179,67,110)(23,180,68,111)(24,181,69,112)(25,182,70,99)(26,169,57,100)(27,170,58,101)(28,171,59,102)(29,48,125,90)(30,49,126,91)(31,50,113,92)(32,51,114,93)(33,52,115,94)(34,53,116,95)(35,54,117,96)(36,55,118,97)(37,56,119,98)(38,43,120,85)(39,44,121,86)(40,45,122,87)(41,46,123,88)(42,47,124,89)(71,137,156,207)(72,138,157,208)(73,139,158,209)(74,140,159,210)(75,127,160,197)(76,128,161,198)(77,129,162,199)(78,130,163,200)(79,131,164,201)(80,132,165,202)(81,133,166,203)(82,134,167,204)(83,135,168,205)(84,136,155,206) );
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,73),(30,74),(31,75),(32,76),(33,77),(34,78),(35,79),(36,80),(37,81),(38,82),(39,83),(40,84),(41,71),(42,72),(43,134),(44,135),(45,136),(46,137),(47,138),(48,139),(49,140),(50,127),(51,128),(52,129),(53,130),(54,131),(55,132),(56,133),(57,143),(58,144),(59,145),(60,146),(61,147),(62,148),(63,149),(64,150),(65,151),(66,152),(67,153),(68,154),(69,141),(70,142),(85,204),(86,205),(87,206),(88,207),(89,208),(90,209),(91,210),(92,197),(93,198),(94,199),(95,200),(96,201),(97,202),(98,203),(99,185),(100,186),(101,187),(102,188),(103,189),(104,190),(105,191),(106,192),(107,193),(108,194),(109,195),(110,196),(111,183),(112,184),(113,160),(114,161),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167),(121,168),(122,155),(123,156),(124,157),(125,158),(126,159),(169,222),(170,223),(171,224),(172,211),(173,212),(174,213),(175,214),(176,215),(177,216),(178,217),(179,218),(180,219),(181,220),(182,221)], [(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)], [(1,86),(2,87),(3,88),(4,89),(5,90),(6,91),(7,92),(8,93),(9,94),(10,95),(11,96),(12,97),(13,98),(14,85),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,197),(22,198),(23,199),(24,200),(25,201),(26,202),(27,203),(28,204),(29,80),(30,81),(31,82),(32,83),(33,84),(34,71),(35,72),(36,73),(37,74),(38,75),(39,76),(40,77),(41,78),(42,79),(43,145),(44,146),(45,147),(46,148),(47,149),(48,150),(49,151),(50,152),(51,153),(52,154),(53,141),(54,142),(55,143),(56,144),(57,132),(58,133),(59,134),(60,135),(61,136),(62,137),(63,138),(64,139),(65,140),(66,127),(67,128),(68,129),(69,130),(70,131),(99,192),(100,193),(101,194),(102,195),(103,196),(104,183),(105,184),(106,185),(107,186),(108,187),(109,188),(110,189),(111,190),(112,191),(113,167),(114,168),(115,155),(116,156),(117,157),(118,158),(119,159),(120,160),(121,161),(122,162),(123,163),(124,164),(125,165),(126,166),(169,215),(170,216),(171,217),(172,218),(173,219),(174,220),(175,221),(176,222),(177,223),(178,224),(179,211),(180,212),(181,213),(182,214)], [(1,198),(2,199),(3,200),(4,201),(5,202),(6,203),(7,204),(8,205),(9,206),(10,207),(11,208),(12,209),(13,210),(14,197),(15,93),(16,94),(17,95),(18,96),(19,97),(20,98),(21,85),(22,86),(23,87),(24,88),(25,89),(26,90),(27,91),(28,92),(29,169),(30,170),(31,171),(32,172),(33,173),(34,174),(35,175),(36,176),(37,177),(38,178),(39,179),(40,180),(41,181),(42,182),(43,66),(44,67),(45,68),(46,69),(47,70),(48,57),(49,58),(50,59),(51,60),(52,61),(53,62),(54,63),(55,64),(56,65),(71,220),(72,221),(73,222),(74,223),(75,224),(76,211),(77,212),(78,213),(79,214),(80,215),(81,216),(82,217),(83,218),(84,219),(99,124),(100,125),(101,126),(102,113),(103,114),(104,115),(105,116),(106,117),(107,118),(108,119),(109,120),(110,121),(111,122),(112,123),(127,145),(128,146),(129,147),(130,148),(131,149),(132,150),(133,151),(134,152),(135,153),(136,154),(137,141),(138,142),(139,143),(140,144),(155,183),(156,184),(157,185),(158,186),(159,187),(160,188),(161,189),(162,190),(163,191),(164,192),(165,193),(166,194),(167,195),(168,196)], [(1,211,146,189),(2,212,147,190),(3,213,148,191),(4,214,149,192),(5,215,150,193),(6,216,151,194),(7,217,152,195),(8,218,153,196),(9,219,154,183),(10,220,141,184),(11,221,142,185),(12,222,143,186),(13,223,144,187),(14,224,145,188),(15,172,60,103),(16,173,61,104),(17,174,62,105),(18,175,63,106),(19,176,64,107),(20,177,65,108),(21,178,66,109),(22,179,67,110),(23,180,68,111),(24,181,69,112),(25,182,70,99),(26,169,57,100),(27,170,58,101),(28,171,59,102),(29,48,125,90),(30,49,126,91),(31,50,113,92),(32,51,114,93),(33,52,115,94),(34,53,116,95),(35,54,117,96),(36,55,118,97),(37,56,119,98),(38,43,120,85),(39,44,121,86),(40,45,122,87),(41,46,123,88),(42,47,124,89),(71,137,156,207),(72,138,157,208),(73,139,158,209),(74,140,159,210),(75,127,160,197),(76,128,161,198),(77,129,162,199),(78,130,163,200),(79,131,164,201),(80,132,165,202),(81,133,166,203),(82,134,167,204),(83,135,168,205),(84,136,155,206)]])
280 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4P | 7A | ··· | 7F | 14A | ··· | 14CL | 14CM | ··· | 14EH | 28A | ··· | 28CR |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
280 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C7 | C14 | C14 | C14 | C28 | D4 | C7×D4 |
| kernel | C22⋊C4×C2×C14 | C14×C22⋊C4 | C23×C28 | C24×C14 | C23×C14 | C22×C22⋊C4 | C2×C22⋊C4 | C23×C4 | C25 | C24 | C22×C14 | C23 |
| # reps | 1 | 12 | 2 | 1 | 16 | 6 | 72 | 12 | 6 | 96 | 8 | 48 |
Matrix representation of C22⋊C4×C2×C14 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 16 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 23 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 23 | 25 |
| 0 | 0 | 0 | 2 | 6 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,23,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,23,2,0,0,0,25,6] >;
C22⋊C4×C2×C14 in GAP, Magma, Sage, TeX
C_2^2\rtimes C_4\times C_2\times C_{14} % in TeX
G:=Group("C2^2:C4xC2xC14"); // GroupNames label
G:=SmallGroup(448,1295);
// by ID
G=gap.SmallGroup(448,1295);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1568,1597]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^14=c^2=d^2=e^4=1,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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations