direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4○D28, C14.4C25, D28⋊14C23, C28.77C24, D14.1C24, C24.73D14, Dic7.2C24, Dic14⋊13C23, (C23×C4)⋊8D7, (C4×D7)⋊8C23, C7⋊D4⋊7C23, C2.5(D7×C24), (C23×C28)⋊11C2, (C2×C28)⋊15C23, (C22×C4)⋊46D14, C4.76(C23×D7), (C22×D28)⋊25C2, (C2×D28)⋊66C22, C22.7(C23×D7), (C2×C14).326C24, (C22×C28)⋊62C22, (C22×Dic14)⋊26C2, (C2×Dic14)⋊77C22, C23.347(C22×D7), (C22×C14).433C23, (C23×C14).116C22, (C2×Dic7).296C23, (C22×D7).245C23, (C23×D7).116C22, (C22×Dic7).239C22, C14⋊1(C2×C4○D4), C7⋊1(C22×C4○D4), (C2×C4×D7)⋊72C22, (D7×C22×C4)⋊26C2, (C2×C4)⋊12(C22×D7), (C2×C14)⋊13(C4○D4), (C2×C7⋊D4)⋊56C22, (C22×C7⋊D4)⋊22C2, SmallGroup(448,1368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22×C4○D28
G = < a,b,c,d,e | a2=b2=c4=e2=1, d14=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d13 >
Subgroups: 3332 in 890 conjugacy classes, 463 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D7, C14, C14, C14, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C22×C4○D4, C2×Dic14, C2×C4×D7, C2×D28, C4○D28, C22×Dic7, C2×C7⋊D4, C22×C28, C22×C28, C23×D7, C23×C14, C22×Dic14, D7×C22×C4, C22×D28, C2×C4○D28, C22×C7⋊D4, C23×C28, C22×C4○D28
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C25, C22×D7, C22×C4○D4, C4○D28, C23×D7, C2×C4○D28, D7×C24, C22×C4○D28
►On 224 points
Generators in S224
(1 136)(2 137)(3 138)(4 139)(5 140)(6 113)(7 114)(8 115)(9 116)(10 117)(11 118)(12 119)(13 120)(14 121)(15 122)(16 123)(17 124)(18 125)(19 126)(20 127)(21 128)(22 129)(23 130)(24 131)(25 132)(26 133)(27 134)(28 135)(29 153)(30 154)(31 155)(32 156)(33 157)(34 158)(35 159)(36 160)(37 161)(38 162)(39 163)(40 164)(41 165)(42 166)(43 167)(44 168)(45 141)(46 142)(47 143)(48 144)(49 145)(50 146)(51 147)(52 148)(53 149)(54 150)(55 151)(56 152)(57 198)(58 199)(59 200)(60 201)(61 202)(62 203)(63 204)(64 205)(65 206)(66 207)(67 208)(68 209)(69 210)(70 211)(71 212)(72 213)(73 214)(74 215)(75 216)(76 217)(77 218)(78 219)(79 220)(80 221)(81 222)(82 223)(83 224)(84 197)(85 196)(86 169)(87 170)(88 171)(89 172)(90 173)(91 174)(92 175)(93 176)(94 177)(95 178)(96 179)(97 180)(98 181)(99 182)(100 183)(101 184)(102 185)(103 186)(104 187)(105 188)(106 189)(107 190)(108 191)(109 192)(110 193)(111 194)(112 195)
(1 186)(2 187)(3 188)(4 189)(5 190)(6 191)(7 192)(8 193)(9 194)(10 195)(11 196)(12 169)(13 170)(14 171)(15 172)(16 173)(17 174)(18 175)(19 176)(20 177)(21 178)(22 179)(23 180)(24 181)(25 182)(26 183)(27 184)(28 185)(29 223)(30 224)(31 197)(32 198)(33 199)(34 200)(35 201)(36 202)(37 203)(38 204)(39 205)(40 206)(41 207)(42 208)(43 209)(44 210)(45 211)(46 212)(47 213)(48 214)(49 215)(50 216)(51 217)(52 218)(53 219)(54 220)(55 221)(56 222)(57 156)(58 157)(59 158)(60 159)(61 160)(62 161)(63 162)(64 163)(65 164)(66 165)(67 166)(68 167)(69 168)(70 141)(71 142)(72 143)(73 144)(74 145)(75 146)(76 147)(77 148)(78 149)(79 150)(80 151)(81 152)(82 153)(83 154)(84 155)(85 118)(86 119)(87 120)(88 121)(89 122)(90 123)(91 124)(92 125)(93 126)(94 127)(95 128)(96 129)(97 130)(98 131)(99 132)(100 133)(101 134)(102 135)(103 136)(104 137)(105 138)(106 139)(107 140)(108 113)(109 114)(110 115)(111 116)(112 117)
(1 154 15 168)(2 155 16 141)(3 156 17 142)(4 157 18 143)(5 158 19 144)(6 159 20 145)(7 160 21 146)(8 161 22 147)(9 162 23 148)(10 163 24 149)(11 164 25 150)(12 165 26 151)(13 166 27 152)(14 167 28 153)(29 121 43 135)(30 122 44 136)(31 123 45 137)(32 124 46 138)(33 125 47 139)(34 126 48 140)(35 127 49 113)(36 128 50 114)(37 129 51 115)(38 130 52 116)(39 131 53 117)(40 132 54 118)(41 133 55 119)(42 134 56 120)(57 174 71 188)(58 175 72 189)(59 176 73 190)(60 177 74 191)(61 178 75 192)(62 179 76 193)(63 180 77 194)(64 181 78 195)(65 182 79 196)(66 183 80 169)(67 184 81 170)(68 185 82 171)(69 186 83 172)(70 187 84 173)(85 206 99 220)(86 207 100 221)(87 208 101 222)(88 209 102 223)(89 210 103 224)(90 211 104 197)(91 212 105 198)(92 213 106 199)(93 214 107 200)(94 215 108 201)(95 216 109 202)(96 217 110 203)(97 218 111 204)(98 219 112 205)
(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)
(1 171)(2 170)(3 169)(4 196)(5 195)(6 194)(7 193)(8 192)(9 191)(10 190)(11 189)(12 188)(13 187)(14 186)(15 185)(16 184)(17 183)(18 182)(19 181)(20 180)(21 179)(22 178)(23 177)(24 176)(25 175)(26 174)(27 173)(28 172)(29 210)(30 209)(31 208)(32 207)(33 206)(34 205)(35 204)(36 203)(37 202)(38 201)(39 200)(40 199)(41 198)(42 197)(43 224)(44 223)(45 222)(46 221)(47 220)(48 219)(49 218)(50 217)(51 216)(52 215)(53 214)(54 213)(55 212)(56 211)(57 165)(58 164)(59 163)(60 162)(61 161)(62 160)(63 159)(64 158)(65 157)(66 156)(67 155)(68 154)(69 153)(70 152)(71 151)(72 150)(73 149)(74 148)(75 147)(76 146)(77 145)(78 144)(79 143)(80 142)(81 141)(82 168)(83 167)(84 166)(85 139)(86 138)(87 137)(88 136)(89 135)(90 134)(91 133)(92 132)(93 131)(94 130)(95 129)(96 128)(97 127)(98 126)(99 125)(100 124)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(112 140)
G:=sub<Sym(224)| (1,136)(2,137)(3,138)(4,139)(5,140)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,198)(58,199)(59,200)(60,201)(61,202)(62,203)(63,204)(64,205)(65,206)(66,207)(67,208)(68,209)(69,210)(70,211)(71,212)(72,213)(73,214)(74,215)(75,216)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,224)(84,197)(85,196)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,181)(99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(109,192)(110,193)(111,194)(112,195), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,182)(26,183)(27,184)(28,185)(29,223)(30,224)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,156)(58,157)(59,158)(60,159)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,113)(109,114)(110,115)(111,116)(112,117), (1,154,15,168)(2,155,16,141)(3,156,17,142)(4,157,18,143)(5,158,19,144)(6,159,20,145)(7,160,21,146)(8,161,22,147)(9,162,23,148)(10,163,24,149)(11,164,25,150)(12,165,26,151)(13,166,27,152)(14,167,28,153)(29,121,43,135)(30,122,44,136)(31,123,45,137)(32,124,46,138)(33,125,47,139)(34,126,48,140)(35,127,49,113)(36,128,50,114)(37,129,51,115)(38,130,52,116)(39,131,53,117)(40,132,54,118)(41,133,55,119)(42,134,56,120)(57,174,71,188)(58,175,72,189)(59,176,73,190)(60,177,74,191)(61,178,75,192)(62,179,76,193)(63,180,77,194)(64,181,78,195)(65,182,79,196)(66,183,80,169)(67,184,81,170)(68,185,82,171)(69,186,83,172)(70,187,84,173)(85,206,99,220)(86,207,100,221)(87,208,101,222)(88,209,102,223)(89,210,103,224)(90,211,104,197)(91,212,105,198)(92,213,106,199)(93,214,107,200)(94,215,108,201)(95,216,109,202)(96,217,110,203)(97,218,111,204)(98,219,112,205), (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), (1,171)(2,170)(3,169)(4,196)(5,195)(6,194)(7,193)(8,192)(9,191)(10,190)(11,189)(12,188)(13,187)(14,186)(15,185)(16,184)(17,183)(18,182)(19,181)(20,180)(21,179)(22,178)(23,177)(24,176)(25,175)(26,174)(27,173)(28,172)(29,210)(30,209)(31,208)(32,207)(33,206)(34,205)(35,204)(36,203)(37,202)(38,201)(39,200)(40,199)(41,198)(42,197)(43,224)(44,223)(45,222)(46,221)(47,220)(48,219)(49,218)(50,217)(51,216)(52,215)(53,214)(54,213)(55,212)(56,211)(57,165)(58,164)(59,163)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141)(82,168)(83,167)(84,166)(85,139)(86,138)(87,137)(88,136)(89,135)(90,134)(91,133)(92,132)(93,131)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(112,140)>;
G:=Group( (1,136)(2,137)(3,138)(4,139)(5,140)(6,113)(7,114)(8,115)(9,116)(10,117)(11,118)(12,119)(13,120)(14,121)(15,122)(16,123)(17,124)(18,125)(19,126)(20,127)(21,128)(22,129)(23,130)(24,131)(25,132)(26,133)(27,134)(28,135)(29,153)(30,154)(31,155)(32,156)(33,157)(34,158)(35,159)(36,160)(37,161)(38,162)(39,163)(40,164)(41,165)(42,166)(43,167)(44,168)(45,141)(46,142)(47,143)(48,144)(49,145)(50,146)(51,147)(52,148)(53,149)(54,150)(55,151)(56,152)(57,198)(58,199)(59,200)(60,201)(61,202)(62,203)(63,204)(64,205)(65,206)(66,207)(67,208)(68,209)(69,210)(70,211)(71,212)(72,213)(73,214)(74,215)(75,216)(76,217)(77,218)(78,219)(79,220)(80,221)(81,222)(82,223)(83,224)(84,197)(85,196)(86,169)(87,170)(88,171)(89,172)(90,173)(91,174)(92,175)(93,176)(94,177)(95,178)(96,179)(97,180)(98,181)(99,182)(100,183)(101,184)(102,185)(103,186)(104,187)(105,188)(106,189)(107,190)(108,191)(109,192)(110,193)(111,194)(112,195), (1,186)(2,187)(3,188)(4,189)(5,190)(6,191)(7,192)(8,193)(9,194)(10,195)(11,196)(12,169)(13,170)(14,171)(15,172)(16,173)(17,174)(18,175)(19,176)(20,177)(21,178)(22,179)(23,180)(24,181)(25,182)(26,183)(27,184)(28,185)(29,223)(30,224)(31,197)(32,198)(33,199)(34,200)(35,201)(36,202)(37,203)(38,204)(39,205)(40,206)(41,207)(42,208)(43,209)(44,210)(45,211)(46,212)(47,213)(48,214)(49,215)(50,216)(51,217)(52,218)(53,219)(54,220)(55,221)(56,222)(57,156)(58,157)(59,158)(60,159)(61,160)(62,161)(63,162)(64,163)(65,164)(66,165)(67,166)(68,167)(69,168)(70,141)(71,142)(72,143)(73,144)(74,145)(75,146)(76,147)(77,148)(78,149)(79,150)(80,151)(81,152)(82,153)(83,154)(84,155)(85,118)(86,119)(87,120)(88,121)(89,122)(90,123)(91,124)(92,125)(93,126)(94,127)(95,128)(96,129)(97,130)(98,131)(99,132)(100,133)(101,134)(102,135)(103,136)(104,137)(105,138)(106,139)(107,140)(108,113)(109,114)(110,115)(111,116)(112,117), (1,154,15,168)(2,155,16,141)(3,156,17,142)(4,157,18,143)(5,158,19,144)(6,159,20,145)(7,160,21,146)(8,161,22,147)(9,162,23,148)(10,163,24,149)(11,164,25,150)(12,165,26,151)(13,166,27,152)(14,167,28,153)(29,121,43,135)(30,122,44,136)(31,123,45,137)(32,124,46,138)(33,125,47,139)(34,126,48,140)(35,127,49,113)(36,128,50,114)(37,129,51,115)(38,130,52,116)(39,131,53,117)(40,132,54,118)(41,133,55,119)(42,134,56,120)(57,174,71,188)(58,175,72,189)(59,176,73,190)(60,177,74,191)(61,178,75,192)(62,179,76,193)(63,180,77,194)(64,181,78,195)(65,182,79,196)(66,183,80,169)(67,184,81,170)(68,185,82,171)(69,186,83,172)(70,187,84,173)(85,206,99,220)(86,207,100,221)(87,208,101,222)(88,209,102,223)(89,210,103,224)(90,211,104,197)(91,212,105,198)(92,213,106,199)(93,214,107,200)(94,215,108,201)(95,216,109,202)(96,217,110,203)(97,218,111,204)(98,219,112,205), (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), (1,171)(2,170)(3,169)(4,196)(5,195)(6,194)(7,193)(8,192)(9,191)(10,190)(11,189)(12,188)(13,187)(14,186)(15,185)(16,184)(17,183)(18,182)(19,181)(20,180)(21,179)(22,178)(23,177)(24,176)(25,175)(26,174)(27,173)(28,172)(29,210)(30,209)(31,208)(32,207)(33,206)(34,205)(35,204)(36,203)(37,202)(38,201)(39,200)(40,199)(41,198)(42,197)(43,224)(44,223)(45,222)(46,221)(47,220)(48,219)(49,218)(50,217)(51,216)(52,215)(53,214)(54,213)(55,212)(56,211)(57,165)(58,164)(59,163)(60,162)(61,161)(62,160)(63,159)(64,158)(65,157)(66,156)(67,155)(68,154)(69,153)(70,152)(71,151)(72,150)(73,149)(74,148)(75,147)(76,146)(77,145)(78,144)(79,143)(80,142)(81,141)(82,168)(83,167)(84,166)(85,139)(86,138)(87,137)(88,136)(89,135)(90,134)(91,133)(92,132)(93,131)(94,130)(95,129)(96,128)(97,127)(98,126)(99,125)(100,124)(101,123)(102,122)(103,121)(104,120)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)(112,140) );
G=PermutationGroup([[(1,136),(2,137),(3,138),(4,139),(5,140),(6,113),(7,114),(8,115),(9,116),(10,117),(11,118),(12,119),(13,120),(14,121),(15,122),(16,123),(17,124),(18,125),(19,126),(20,127),(21,128),(22,129),(23,130),(24,131),(25,132),(26,133),(27,134),(28,135),(29,153),(30,154),(31,155),(32,156),(33,157),(34,158),(35,159),(36,160),(37,161),(38,162),(39,163),(40,164),(41,165),(42,166),(43,167),(44,168),(45,141),(46,142),(47,143),(48,144),(49,145),(50,146),(51,147),(52,148),(53,149),(54,150),(55,151),(56,152),(57,198),(58,199),(59,200),(60,201),(61,202),(62,203),(63,204),(64,205),(65,206),(66,207),(67,208),(68,209),(69,210),(70,211),(71,212),(72,213),(73,214),(74,215),(75,216),(76,217),(77,218),(78,219),(79,220),(80,221),(81,222),(82,223),(83,224),(84,197),(85,196),(86,169),(87,170),(88,171),(89,172),(90,173),(91,174),(92,175),(93,176),(94,177),(95,178),(96,179),(97,180),(98,181),(99,182),(100,183),(101,184),(102,185),(103,186),(104,187),(105,188),(106,189),(107,190),(108,191),(109,192),(110,193),(111,194),(112,195)], [(1,186),(2,187),(3,188),(4,189),(5,190),(6,191),(7,192),(8,193),(9,194),(10,195),(11,196),(12,169),(13,170),(14,171),(15,172),(16,173),(17,174),(18,175),(19,176),(20,177),(21,178),(22,179),(23,180),(24,181),(25,182),(26,183),(27,184),(28,185),(29,223),(30,224),(31,197),(32,198),(33,199),(34,200),(35,201),(36,202),(37,203),(38,204),(39,205),(40,206),(41,207),(42,208),(43,209),(44,210),(45,211),(46,212),(47,213),(48,214),(49,215),(50,216),(51,217),(52,218),(53,219),(54,220),(55,221),(56,222),(57,156),(58,157),(59,158),(60,159),(61,160),(62,161),(63,162),(64,163),(65,164),(66,165),(67,166),(68,167),(69,168),(70,141),(71,142),(72,143),(73,144),(74,145),(75,146),(76,147),(77,148),(78,149),(79,150),(80,151),(81,152),(82,153),(83,154),(84,155),(85,118),(86,119),(87,120),(88,121),(89,122),(90,123),(91,124),(92,125),(93,126),(94,127),(95,128),(96,129),(97,130),(98,131),(99,132),(100,133),(101,134),(102,135),(103,136),(104,137),(105,138),(106,139),(107,140),(108,113),(109,114),(110,115),(111,116),(112,117)], [(1,154,15,168),(2,155,16,141),(3,156,17,142),(4,157,18,143),(5,158,19,144),(6,159,20,145),(7,160,21,146),(8,161,22,147),(9,162,23,148),(10,163,24,149),(11,164,25,150),(12,165,26,151),(13,166,27,152),(14,167,28,153),(29,121,43,135),(30,122,44,136),(31,123,45,137),(32,124,46,138),(33,125,47,139),(34,126,48,140),(35,127,49,113),(36,128,50,114),(37,129,51,115),(38,130,52,116),(39,131,53,117),(40,132,54,118),(41,133,55,119),(42,134,56,120),(57,174,71,188),(58,175,72,189),(59,176,73,190),(60,177,74,191),(61,178,75,192),(62,179,76,193),(63,180,77,194),(64,181,78,195),(65,182,79,196),(66,183,80,169),(67,184,81,170),(68,185,82,171),(69,186,83,172),(70,187,84,173),(85,206,99,220),(86,207,100,221),(87,208,101,222),(88,209,102,223),(89,210,103,224),(90,211,104,197),(91,212,105,198),(92,213,106,199),(93,214,107,200),(94,215,108,201),(95,216,109,202),(96,217,110,203),(97,218,111,204),(98,219,112,205)], [(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)], [(1,171),(2,170),(3,169),(4,196),(5,195),(6,194),(7,193),(8,192),(9,191),(10,190),(11,189),(12,188),(13,187),(14,186),(15,185),(16,184),(17,183),(18,182),(19,181),(20,180),(21,179),(22,178),(23,177),(24,176),(25,175),(26,174),(27,173),(28,172),(29,210),(30,209),(31,208),(32,207),(33,206),(34,205),(35,204),(36,203),(37,202),(38,201),(39,200),(40,199),(41,198),(42,197),(43,224),(44,223),(45,222),(46,221),(47,220),(48,219),(49,218),(50,217),(51,216),(52,215),(53,214),(54,213),(55,212),(56,211),(57,165),(58,164),(59,163),(60,162),(61,161),(62,160),(63,159),(64,158),(65,157),(66,156),(67,155),(68,154),(69,153),(70,152),(71,151),(72,150),(73,149),(74,148),(75,147),(76,146),(77,145),(78,144),(79,143),(80,142),(81,141),(82,168),(83,167),(84,166),(85,139),(86,138),(87,137),(88,136),(89,135),(90,134),(91,133),(92,132),(93,131),(94,130),(95,129),(96,128),(97,127),(98,126),(99,125),(100,124),(101,123),(102,122),(103,121),(104,120),(105,119),(106,118),(107,117),(108,116),(109,115),(110,114),(111,113),(112,140)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4T | 7A | 7B | 7C | 14A | ··· | 14AS | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D7 | C4○D4 | D14 | D14 | C4○D28 |
| kernel | C22×C4○D28 | C22×Dic14 | D7×C22×C4 | C22×D28 | C2×C4○D28 | C22×C7⋊D4 | C23×C28 | C23×C4 | C2×C14 | C22×C4 | C24 | C22 |
| # reps | 1 | 1 | 2 | 1 | 24 | 2 | 1 | 3 | 8 | 42 | 3 | 48 |
Matrix representation of C22×C4○D28 ►in GL5(𝔽29)
| 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 21 |
| 0 | 0 | 0 | 16 | 20 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 28 |
| 0 | 0 | 0 | 5 | 21 |
G:=sub<GL(5,GF(29))| [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],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,12,0,0,0,0,0,0,27,16,0,0,0,21,20],[1,0,0,0,0,0,0,12,0,0,0,17,0,0,0,0,0,0,8,5,0,0,0,28,21] >;
C22×C4○D28 in GAP, Magma, Sage, TeX
C_2^2\times C_4\circ D_{28} % in TeX
G:=Group("C2^2xC4oD28"); // GroupNames label
G:=SmallGroup(448,1368);
// by ID
G=gap.SmallGroup(448,1368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,136,1684,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^14=c^2,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,c*e=e*c,e*d*e=c^2*d^13>;
// generators/relations