direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C22⋊C4×C2×C14, C24⋊5C28, C25.3C14, (C23×C4)⋊4C14, (C23×C14)⋊6C4, (C23×C28)⋊5C2, C23⋊6(C2×C28), (C2×C28)⋊13C23, (C24×C14).2C2, C2.1(C23×C28), C23.58(C7×D4), C24.31(C2×C14), C14.53(C23×C4), C22⋊2(C22×C28), C22.57(D4×C14), (C2×C14).332C24, (C22×C28)⋊57C22, C14.177(C22×D4), (C22×C14).219D4, 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×C14)⋊16(C2×C4), (C22×C4)⋊15(C2×C14), (C2×C14).679(C2×D4), SmallGroup(448,1295)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1010 in 674 conjugacy classes, 338 normal (12 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×8], C22, C22 [×42], C22 [×56], C7, C2×C4 [×8], C2×C4 [×24], C23 [×43], C23 [×56], C14, C14 [×14], C14 [×8], C22⋊C4 [×16], C22×C4 [×12], C22×C4 [×8], C24, C24 [×14], C24 [×8], C28 [×8], C2×C14, C2×C14 [×42], C2×C14 [×56], C2×C22⋊C4 [×12], C23×C4 [×2], C25, C2×C28 [×8], C2×C28 [×24], C22×C14 [×43], C22×C14 [×56], C22×C22⋊C4, C7×C22⋊C4 [×16], C22×C28 [×12], C22×C28 [×8], C23×C14, C23×C14 [×14], C23×C14 [×8], C14×C22⋊C4 [×12], C23×C28 [×2], C24×C14, C22⋊C4×C2×C14
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C7, C2×C4 [×28], D4 [×8], C23 [×15], C14 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C28 [×8], C2×C14 [×35], C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C2×C28 [×28], C7×D4 [×8], C22×C14 [×15], C22×C22⋊C4, C7×C22⋊C4 [×16], C22×C28 [×14], D4×C14 [×12], C23×C14, C14×C22⋊C4 [×12], C23×C28, D4×C2×C14 [×2], C22⋊C4×C2×C14
Generators and relations
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 >
►On 224 points
Generators in S224
(1 83)(2 84)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 81)(14 82)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 85)(27 86)(28 87)(29 133)(30 134)(31 135)(32 136)(33 137)(34 138)(35 139)(36 140)(37 127)(38 128)(39 129)(40 130)(41 131)(42 132)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 99)(64 100)(65 101)(66 102)(67 103)(68 104)(69 105)(70 106)(113 162)(114 163)(115 164)(116 165)(117 166)(118 167)(119 168)(120 155)(121 156)(122 157)(123 158)(124 159)(125 160)(126 161)(169 186)(170 187)(171 188)(172 189)(173 190)(174 191)(175 192)(176 193)(177 194)(178 195)(179 196)(180 183)(181 184)(182 185)(197 222)(198 223)(199 224)(200 211)(201 212)(202 213)(203 214)(204 215)(205 216)(206 217)(207 218)(208 219)(209 220)(210 221)
(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)
(1 193)(2 194)(3 195)(4 196)(5 183)(6 184)(7 185)(8 186)(9 187)(10 188)(11 189)(12 190)(13 191)(14 192)(15 95)(16 96)(17 97)(18 98)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 91)(26 92)(27 93)(28 94)(29 125)(30 126)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 121)(40 122)(41 123)(42 124)(43 142)(44 143)(45 144)(46 145)(47 146)(48 147)(49 148)(50 149)(51 150)(52 151)(53 152)(54 153)(55 154)(56 141)(57 100)(58 101)(59 102)(60 103)(61 104)(62 105)(63 106)(64 107)(65 108)(66 109)(67 110)(68 111)(69 112)(70 99)(71 178)(72 179)(73 180)(74 181)(75 182)(76 169)(77 170)(78 171)(79 172)(80 173)(81 174)(82 175)(83 176)(84 177)(127 168)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 161)(135 162)(136 163)(137 164)(138 165)(139 166)(140 167)(197 215)(198 216)(199 217)(200 218)(201 219)(202 220)(203 221)(204 222)(205 223)(206 224)(207 211)(208 212)(209 213)(210 214)
(1 169)(2 170)(3 171)(4 172)(5 173)(6 174)(7 175)(8 176)(9 177)(10 178)(11 179)(12 180)(13 181)(14 182)(15 56)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 167)(30 168)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 161)(38 162)(39 163)(40 164)(41 165)(42 166)(57 211)(58 212)(59 213)(60 214)(61 215)(62 216)(63 217)(64 218)(65 219)(66 220)(67 221)(68 222)(69 223)(70 224)(71 188)(72 189)(73 190)(74 191)(75 192)(76 193)(77 194)(78 195)(79 196)(80 183)(81 184)(82 185)(83 186)(84 187)(85 145)(86 146)(87 147)(88 148)(89 149)(90 150)(91 151)(92 152)(93 153)(94 154)(95 141)(96 142)(97 143)(98 144)(99 206)(100 207)(101 208)(102 209)(103 210)(104 197)(105 198)(106 199)(107 200)(108 201)(109 202)(110 203)(111 204)(112 205)(113 128)(114 129)(115 130)(116 131)(117 132)(118 133)(119 134)(120 135)(121 136)(122 137)(123 138)(124 139)(125 140)(126 127)
(1 96 159 104)(2 97 160 105)(3 98 161 106)(4 85 162 107)(5 86 163 108)(6 87 164 109)(7 88 165 110)(8 89 166 111)(9 90 167 112)(10 91 168 99)(11 92 155 100)(12 93 156 101)(13 94 157 102)(14 95 158 103)(15 116 60 75)(16 117 61 76)(17 118 62 77)(18 119 63 78)(19 120 64 79)(20 121 65 80)(21 122 66 81)(22 123 67 82)(23 124 68 83)(24 125 69 84)(25 126 70 71)(26 113 57 72)(27 114 58 73)(28 115 59 74)(29 205 177 150)(30 206 178 151)(31 207 179 152)(32 208 180 153)(33 209 181 154)(34 210 182 141)(35 197 169 142)(36 198 170 143)(37 199 171 144)(38 200 172 145)(39 201 173 146)(40 202 174 147)(41 203 175 148)(42 204 176 149)(43 132 215 193)(44 133 216 194)(45 134 217 195)(46 135 218 196)(47 136 219 183)(48 137 220 184)(49 138 221 185)(50 139 222 186)(51 140 223 187)(52 127 224 188)(53 128 211 189)(54 129 212 190)(55 130 213 191)(56 131 214 192)
G:=sub<Sym(224)| (1,83)(2,84)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,85)(27,86)(28,87)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(169,186)(170,187)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,183)(181,184)(182,185)(197,222)(198,223)(199,224)(200,211)(201,212)(202,213)(203,214)(204,215)(205,216)(206,217)(207,218)(208,219)(209,220)(210,221), (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), (1,193)(2,194)(3,195)(4,196)(5,183)(6,184)(7,185)(8,186)(9,187)(10,188)(11,189)(12,190)(13,191)(14,192)(15,95)(16,96)(17,97)(18,98)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,125)(30,126)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,141)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,99)(71,178)(72,179)(73,180)(74,181)(75,182)(76,169)(77,170)(78,171)(79,172)(80,173)(81,174)(82,175)(83,176)(84,177)(127,168)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(197,215)(198,216)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222)(205,223)(206,224)(207,211)(208,212)(209,213)(210,214), (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,56)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,167)(30,168)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(57,211)(58,212)(59,213)(60,214)(61,215)(62,216)(63,217)(64,218)(65,219)(66,220)(67,221)(68,222)(69,223)(70,224)(71,188)(72,189)(73,190)(74,191)(75,192)(76,193)(77,194)(78,195)(79,196)(80,183)(81,184)(82,185)(83,186)(84,187)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(99,206)(100,207)(101,208)(102,209)(103,210)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(121,136)(122,137)(123,138)(124,139)(125,140)(126,127), (1,96,159,104)(2,97,160,105)(3,98,161,106)(4,85,162,107)(5,86,163,108)(6,87,164,109)(7,88,165,110)(8,89,166,111)(9,90,167,112)(10,91,168,99)(11,92,155,100)(12,93,156,101)(13,94,157,102)(14,95,158,103)(15,116,60,75)(16,117,61,76)(17,118,62,77)(18,119,63,78)(19,120,64,79)(20,121,65,80)(21,122,66,81)(22,123,67,82)(23,124,68,83)(24,125,69,84)(25,126,70,71)(26,113,57,72)(27,114,58,73)(28,115,59,74)(29,205,177,150)(30,206,178,151)(31,207,179,152)(32,208,180,153)(33,209,181,154)(34,210,182,141)(35,197,169,142)(36,198,170,143)(37,199,171,144)(38,200,172,145)(39,201,173,146)(40,202,174,147)(41,203,175,148)(42,204,176,149)(43,132,215,193)(44,133,216,194)(45,134,217,195)(46,135,218,196)(47,136,219,183)(48,137,220,184)(49,138,221,185)(50,139,222,186)(51,140,223,187)(52,127,224,188)(53,128,211,189)(54,129,212,190)(55,130,213,191)(56,131,214,192)>;
G:=Group( (1,83)(2,84)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,81)(14,82)(15,88)(16,89)(17,90)(18,91)(19,92)(20,93)(21,94)(22,95)(23,96)(24,97)(25,98)(26,85)(27,86)(28,87)(29,133)(30,134)(31,135)(32,136)(33,137)(34,138)(35,139)(36,140)(37,127)(38,128)(39,129)(40,130)(41,131)(42,132)(43,149)(44,150)(45,151)(46,152)(47,153)(48,154)(49,141)(50,142)(51,143)(52,144)(53,145)(54,146)(55,147)(56,148)(57,107)(58,108)(59,109)(60,110)(61,111)(62,112)(63,99)(64,100)(65,101)(66,102)(67,103)(68,104)(69,105)(70,106)(113,162)(114,163)(115,164)(116,165)(117,166)(118,167)(119,168)(120,155)(121,156)(122,157)(123,158)(124,159)(125,160)(126,161)(169,186)(170,187)(171,188)(172,189)(173,190)(174,191)(175,192)(176,193)(177,194)(178,195)(179,196)(180,183)(181,184)(182,185)(197,222)(198,223)(199,224)(200,211)(201,212)(202,213)(203,214)(204,215)(205,216)(206,217)(207,218)(208,219)(209,220)(210,221), (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), (1,193)(2,194)(3,195)(4,196)(5,183)(6,184)(7,185)(8,186)(9,187)(10,188)(11,189)(12,190)(13,191)(14,192)(15,95)(16,96)(17,97)(18,98)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,91)(26,92)(27,93)(28,94)(29,125)(30,126)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,121)(40,122)(41,123)(42,124)(43,142)(44,143)(45,144)(46,145)(47,146)(48,147)(49,148)(50,149)(51,150)(52,151)(53,152)(54,153)(55,154)(56,141)(57,100)(58,101)(59,102)(60,103)(61,104)(62,105)(63,106)(64,107)(65,108)(66,109)(67,110)(68,111)(69,112)(70,99)(71,178)(72,179)(73,180)(74,181)(75,182)(76,169)(77,170)(78,171)(79,172)(80,173)(81,174)(82,175)(83,176)(84,177)(127,168)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(197,215)(198,216)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222)(205,223)(206,224)(207,211)(208,212)(209,213)(210,214), (1,169)(2,170)(3,171)(4,172)(5,173)(6,174)(7,175)(8,176)(9,177)(10,178)(11,179)(12,180)(13,181)(14,182)(15,56)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,167)(30,168)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(57,211)(58,212)(59,213)(60,214)(61,215)(62,216)(63,217)(64,218)(65,219)(66,220)(67,221)(68,222)(69,223)(70,224)(71,188)(72,189)(73,190)(74,191)(75,192)(76,193)(77,194)(78,195)(79,196)(80,183)(81,184)(82,185)(83,186)(84,187)(85,145)(86,146)(87,147)(88,148)(89,149)(90,150)(91,151)(92,152)(93,153)(94,154)(95,141)(96,142)(97,143)(98,144)(99,206)(100,207)(101,208)(102,209)(103,210)(104,197)(105,198)(106,199)(107,200)(108,201)(109,202)(110,203)(111,204)(112,205)(113,128)(114,129)(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(121,136)(122,137)(123,138)(124,139)(125,140)(126,127), (1,96,159,104)(2,97,160,105)(3,98,161,106)(4,85,162,107)(5,86,163,108)(6,87,164,109)(7,88,165,110)(8,89,166,111)(9,90,167,112)(10,91,168,99)(11,92,155,100)(12,93,156,101)(13,94,157,102)(14,95,158,103)(15,116,60,75)(16,117,61,76)(17,118,62,77)(18,119,63,78)(19,120,64,79)(20,121,65,80)(21,122,66,81)(22,123,67,82)(23,124,68,83)(24,125,69,84)(25,126,70,71)(26,113,57,72)(27,114,58,73)(28,115,59,74)(29,205,177,150)(30,206,178,151)(31,207,179,152)(32,208,180,153)(33,209,181,154)(34,210,182,141)(35,197,169,142)(36,198,170,143)(37,199,171,144)(38,200,172,145)(39,201,173,146)(40,202,174,147)(41,203,175,148)(42,204,176,149)(43,132,215,193)(44,133,216,194)(45,134,217,195)(46,135,218,196)(47,136,219,183)(48,137,220,184)(49,138,221,185)(50,139,222,186)(51,140,223,187)(52,127,224,188)(53,128,211,189)(54,129,212,190)(55,130,213,191)(56,131,214,192) );
G=PermutationGroup([(1,83),(2,84),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,81),(14,82),(15,88),(16,89),(17,90),(18,91),(19,92),(20,93),(21,94),(22,95),(23,96),(24,97),(25,98),(26,85),(27,86),(28,87),(29,133),(30,134),(31,135),(32,136),(33,137),(34,138),(35,139),(36,140),(37,127),(38,128),(39,129),(40,130),(41,131),(42,132),(43,149),(44,150),(45,151),(46,152),(47,153),(48,154),(49,141),(50,142),(51,143),(52,144),(53,145),(54,146),(55,147),(56,148),(57,107),(58,108),(59,109),(60,110),(61,111),(62,112),(63,99),(64,100),(65,101),(66,102),(67,103),(68,104),(69,105),(70,106),(113,162),(114,163),(115,164),(116,165),(117,166),(118,167),(119,168),(120,155),(121,156),(122,157),(123,158),(124,159),(125,160),(126,161),(169,186),(170,187),(171,188),(172,189),(173,190),(174,191),(175,192),(176,193),(177,194),(178,195),(179,196),(180,183),(181,184),(182,185),(197,222),(198,223),(199,224),(200,211),(201,212),(202,213),(203,214),(204,215),(205,216),(206,217),(207,218),(208,219),(209,220),(210,221)], [(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)], [(1,193),(2,194),(3,195),(4,196),(5,183),(6,184),(7,185),(8,186),(9,187),(10,188),(11,189),(12,190),(13,191),(14,192),(15,95),(16,96),(17,97),(18,98),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,91),(26,92),(27,93),(28,94),(29,125),(30,126),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,121),(40,122),(41,123),(42,124),(43,142),(44,143),(45,144),(46,145),(47,146),(48,147),(49,148),(50,149),(51,150),(52,151),(53,152),(54,153),(55,154),(56,141),(57,100),(58,101),(59,102),(60,103),(61,104),(62,105),(63,106),(64,107),(65,108),(66,109),(67,110),(68,111),(69,112),(70,99),(71,178),(72,179),(73,180),(74,181),(75,182),(76,169),(77,170),(78,171),(79,172),(80,173),(81,174),(82,175),(83,176),(84,177),(127,168),(128,155),(129,156),(130,157),(131,158),(132,159),(133,160),(134,161),(135,162),(136,163),(137,164),(138,165),(139,166),(140,167),(197,215),(198,216),(199,217),(200,218),(201,219),(202,220),(203,221),(204,222),(205,223),(206,224),(207,211),(208,212),(209,213),(210,214)], [(1,169),(2,170),(3,171),(4,172),(5,173),(6,174),(7,175),(8,176),(9,177),(10,178),(11,179),(12,180),(13,181),(14,182),(15,56),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,167),(30,168),(31,155),(32,156),(33,157),(34,158),(35,159),(36,160),(37,161),(38,162),(39,163),(40,164),(41,165),(42,166),(57,211),(58,212),(59,213),(60,214),(61,215),(62,216),(63,217),(64,218),(65,219),(66,220),(67,221),(68,222),(69,223),(70,224),(71,188),(72,189),(73,190),(74,191),(75,192),(76,193),(77,194),(78,195),(79,196),(80,183),(81,184),(82,185),(83,186),(84,187),(85,145),(86,146),(87,147),(88,148),(89,149),(90,150),(91,151),(92,152),(93,153),(94,154),(95,141),(96,142),(97,143),(98,144),(99,206),(100,207),(101,208),(102,209),(103,210),(104,197),(105,198),(106,199),(107,200),(108,201),(109,202),(110,203),(111,204),(112,205),(113,128),(114,129),(115,130),(116,131),(117,132),(118,133),(119,134),(120,135),(121,136),(122,137),(123,138),(124,139),(125,140),(126,127)], [(1,96,159,104),(2,97,160,105),(3,98,161,106),(4,85,162,107),(5,86,163,108),(6,87,164,109),(7,88,165,110),(8,89,166,111),(9,90,167,112),(10,91,168,99),(11,92,155,100),(12,93,156,101),(13,94,157,102),(14,95,158,103),(15,116,60,75),(16,117,61,76),(17,118,62,77),(18,119,63,78),(19,120,64,79),(20,121,65,80),(21,122,66,81),(22,123,67,82),(23,124,68,83),(24,125,69,84),(25,126,70,71),(26,113,57,72),(27,114,58,73),(28,115,59,74),(29,205,177,150),(30,206,178,151),(31,207,179,152),(32,208,180,153),(33,209,181,154),(34,210,182,141),(35,197,169,142),(36,198,170,143),(37,199,171,144),(38,200,172,145),(39,201,173,146),(40,202,174,147),(41,203,175,148),(42,204,176,149),(43,132,215,193),(44,133,216,194),(45,134,217,195),(46,135,218,196),(47,136,219,183),(48,137,220,184),(49,138,221,185),(50,139,222,186),(51,140,223,187),(52,127,224,188),(53,128,211,189),(54,129,212,190),(55,130,213,191),(56,131,214,192)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀