metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.22D28, (C2×C56)⋊10C4, (C2×C8)⋊6Dic7, C56.73(C2×C4), C56⋊1C4⋊29C2, C8⋊Dic7⋊29C2, C28.47(C4⋊C4), C28.75(C2×Q8), (C2×C28).61Q8, (C2×C4).169D28, (C2×C8).307D14, (C2×C28).401D4, C8.17(C2×Dic7), (C22×C8).11D7, C14.15(C4○D8), (C22×C56).17C2, C4.23(C4⋊Dic7), (C2×C4).50Dic14, C4.41(C2×Dic14), C22.52(C2×D28), C2.4(D56⋊7C2), C28.171(C22×C4), (C2×C56).392C22, (C2×C28).765C23, (C22×C4).425D14, (C22×C14).137D4, C7⋊4(C23.25D4), C4.25(C22×Dic7), C4⋊Dic7.281C22, C22.13(C4⋊Dic7), (C22×C28).539C22, C23.21D14.5C2, C14.46(C2×C4⋊C4), C2.12(C2×C4⋊Dic7), (C2×C14).51(C4⋊C4), (C2×C28).306(C2×C4), (C2×C14).155(C2×D4), (C2×C4).83(C2×Dic7), (C2×C4).712(C22×D7), SmallGroup(448,640)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.22D28
G = < a,b,c,d,e | a2=b2=c2=1, d28=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d27 >
Subgroups: 420 in 114 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic7, C28, C28, C2×C14, C2×C14, C2×C14, C4.Q8, C2.D8, C42⋊C2, C22×C8, C56, C2×Dic7, C2×C28, C2×C28, C22×C14, C23.25D4, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C2×C56, C22×C28, C8⋊Dic7, C56⋊1C4, C23.21D14, C22×C56, C23.22D28
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic7, D14, C2×C4⋊C4, C4○D8, Dic14, D28, C2×Dic7, C22×D7, C23.25D4, C4⋊Dic7, C2×Dic14, C2×D28, C22×Dic7, D56⋊7C2, C2×C4⋊Dic7, C23.22D28
►On 224 points
Generators in S224
(1 191)(2 192)(3 193)(4 194)(5 195)(6 196)(7 197)(8 198)(9 199)(10 200)(11 201)(12 202)(13 203)(14 204)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 220)(31 221)(32 222)(33 223)(34 224)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 139)(58 140)(59 141)(60 142)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 161)(80 162)(81 163)(82 164)(83 165)(84 166)(85 167)(86 168)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 121)(96 122)(97 123)(98 124)(99 125)(100 126)(101 127)(102 128)(103 129)(104 130)(105 131)(106 132)(107 133)(108 134)(109 135)(110 136)(111 137)(112 138)
(1 191)(2 192)(3 193)(4 194)(5 195)(6 196)(7 197)(8 198)(9 199)(10 200)(11 201)(12 202)(13 203)(14 204)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 211)(22 212)(23 213)(24 214)(25 215)(26 216)(27 217)(28 218)(29 219)(30 220)(31 221)(32 222)(33 223)(34 224)(35 169)(36 170)(37 171)(38 172)(39 173)(40 174)(41 175)(42 176)(43 177)(44 178)(45 179)(46 180)(47 181)(48 182)(49 183)(50 184)(51 185)(52 186)(53 187)(54 188)(55 189)(56 190)(57 167)(58 168)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)(73 127)(74 128)(75 129)(76 130)(77 131)(78 132)(79 133)(80 134)(81 135)(82 136)(83 137)(84 138)(85 139)(86 140)(87 141)(88 142)(89 143)(90 144)(91 145)(92 146)(93 147)(94 148)(95 149)(96 150)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(169 197)(170 198)(171 199)(172 200)(173 201)(174 202)(175 203)(176 204)(177 205)(178 206)(179 207)(180 208)(181 209)(182 210)(183 211)(184 212)(185 213)(186 214)(187 215)(188 216)(189 217)(190 218)(191 219)(192 220)(193 221)(194 222)(195 223)(196 224)
(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 70 219 152)(2 97 220 123)(3 68 221 150)(4 95 222 121)(5 66 223 148)(6 93 224 119)(7 64 169 146)(8 91 170 117)(9 62 171 144)(10 89 172 115)(11 60 173 142)(12 87 174 113)(13 58 175 140)(14 85 176 167)(15 112 177 138)(16 83 178 165)(17 110 179 136)(18 81 180 163)(19 108 181 134)(20 79 182 161)(21 106 183 132)(22 77 184 159)(23 104 185 130)(24 75 186 157)(25 102 187 128)(26 73 188 155)(27 100 189 126)(28 71 190 153)(29 98 191 124)(30 69 192 151)(31 96 193 122)(32 67 194 149)(33 94 195 120)(34 65 196 147)(35 92 197 118)(36 63 198 145)(37 90 199 116)(38 61 200 143)(39 88 201 114)(40 59 202 141)(41 86 203 168)(42 57 204 139)(43 84 205 166)(44 111 206 137)(45 82 207 164)(46 109 208 135)(47 80 209 162)(48 107 210 133)(49 78 211 160)(50 105 212 131)(51 76 213 158)(52 103 214 129)(53 74 215 156)(54 101 216 127)(55 72 217 154)(56 99 218 125)
G:=sub<Sym(224)| (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,197)(8,198)(9,199)(10,200)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,218)(29,219)(30,220)(31,221)(32,222)(33,223)(34,224)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,161)(80,162)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138), (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,197)(8,198)(9,199)(10,200)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,218)(29,219)(30,220)(31,221)(32,222)(33,223)(34,224)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,167)(58,168)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,70,219,152)(2,97,220,123)(3,68,221,150)(4,95,222,121)(5,66,223,148)(6,93,224,119)(7,64,169,146)(8,91,170,117)(9,62,171,144)(10,89,172,115)(11,60,173,142)(12,87,174,113)(13,58,175,140)(14,85,176,167)(15,112,177,138)(16,83,178,165)(17,110,179,136)(18,81,180,163)(19,108,181,134)(20,79,182,161)(21,106,183,132)(22,77,184,159)(23,104,185,130)(24,75,186,157)(25,102,187,128)(26,73,188,155)(27,100,189,126)(28,71,190,153)(29,98,191,124)(30,69,192,151)(31,96,193,122)(32,67,194,149)(33,94,195,120)(34,65,196,147)(35,92,197,118)(36,63,198,145)(37,90,199,116)(38,61,200,143)(39,88,201,114)(40,59,202,141)(41,86,203,168)(42,57,204,139)(43,84,205,166)(44,111,206,137)(45,82,207,164)(46,109,208,135)(47,80,209,162)(48,107,210,133)(49,78,211,160)(50,105,212,131)(51,76,213,158)(52,103,214,129)(53,74,215,156)(54,101,216,127)(55,72,217,154)(56,99,218,125)>;
G:=Group( (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,197)(8,198)(9,199)(10,200)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,218)(29,219)(30,220)(31,221)(32,222)(33,223)(34,224)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,139)(58,140)(59,141)(60,142)(61,143)(62,144)(63,145)(64,146)(65,147)(66,148)(67,149)(68,150)(69,151)(70,152)(71,153)(72,154)(73,155)(74,156)(75,157)(76,158)(77,159)(78,160)(79,161)(80,162)(81,163)(82,164)(83,165)(84,166)(85,167)(86,168)(87,113)(88,114)(89,115)(90,116)(91,117)(92,118)(93,119)(94,120)(95,121)(96,122)(97,123)(98,124)(99,125)(100,126)(101,127)(102,128)(103,129)(104,130)(105,131)(106,132)(107,133)(108,134)(109,135)(110,136)(111,137)(112,138), (1,191)(2,192)(3,193)(4,194)(5,195)(6,196)(7,197)(8,198)(9,199)(10,200)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,211)(22,212)(23,213)(24,214)(25,215)(26,216)(27,217)(28,218)(29,219)(30,220)(31,221)(32,222)(33,223)(34,224)(35,169)(36,170)(37,171)(38,172)(39,173)(40,174)(41,175)(42,176)(43,177)(44,178)(45,179)(46,180)(47,181)(48,182)(49,183)(50,184)(51,185)(52,186)(53,187)(54,188)(55,189)(56,190)(57,167)(58,168)(59,113)(60,114)(61,115)(62,116)(63,117)(64,118)(65,119)(66,120)(67,121)(68,122)(69,123)(70,124)(71,125)(72,126)(73,127)(74,128)(75,129)(76,130)(77,131)(78,132)(79,133)(80,134)(81,135)(82,136)(83,137)(84,138)(85,139)(86,140)(87,141)(88,142)(89,143)(90,144)(91,145)(92,146)(93,147)(94,148)(95,149)(96,150)(97,151)(98,152)(99,153)(100,154)(101,155)(102,156)(103,157)(104,158)(105,159)(106,160)(107,161)(108,162)(109,163)(110,164)(111,165)(112,166), (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(169,197)(170,198)(171,199)(172,200)(173,201)(174,202)(175,203)(176,204)(177,205)(178,206)(179,207)(180,208)(181,209)(182,210)(183,211)(184,212)(185,213)(186,214)(187,215)(188,216)(189,217)(190,218)(191,219)(192,220)(193,221)(194,222)(195,223)(196,224), (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), (1,70,219,152)(2,97,220,123)(3,68,221,150)(4,95,222,121)(5,66,223,148)(6,93,224,119)(7,64,169,146)(8,91,170,117)(9,62,171,144)(10,89,172,115)(11,60,173,142)(12,87,174,113)(13,58,175,140)(14,85,176,167)(15,112,177,138)(16,83,178,165)(17,110,179,136)(18,81,180,163)(19,108,181,134)(20,79,182,161)(21,106,183,132)(22,77,184,159)(23,104,185,130)(24,75,186,157)(25,102,187,128)(26,73,188,155)(27,100,189,126)(28,71,190,153)(29,98,191,124)(30,69,192,151)(31,96,193,122)(32,67,194,149)(33,94,195,120)(34,65,196,147)(35,92,197,118)(36,63,198,145)(37,90,199,116)(38,61,200,143)(39,88,201,114)(40,59,202,141)(41,86,203,168)(42,57,204,139)(43,84,205,166)(44,111,206,137)(45,82,207,164)(46,109,208,135)(47,80,209,162)(48,107,210,133)(49,78,211,160)(50,105,212,131)(51,76,213,158)(52,103,214,129)(53,74,215,156)(54,101,216,127)(55,72,217,154)(56,99,218,125) );
G=PermutationGroup([[(1,191),(2,192),(3,193),(4,194),(5,195),(6,196),(7,197),(8,198),(9,199),(10,200),(11,201),(12,202),(13,203),(14,204),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,211),(22,212),(23,213),(24,214),(25,215),(26,216),(27,217),(28,218),(29,219),(30,220),(31,221),(32,222),(33,223),(34,224),(35,169),(36,170),(37,171),(38,172),(39,173),(40,174),(41,175),(42,176),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,183),(50,184),(51,185),(52,186),(53,187),(54,188),(55,189),(56,190),(57,139),(58,140),(59,141),(60,142),(61,143),(62,144),(63,145),(64,146),(65,147),(66,148),(67,149),(68,150),(69,151),(70,152),(71,153),(72,154),(73,155),(74,156),(75,157),(76,158),(77,159),(78,160),(79,161),(80,162),(81,163),(82,164),(83,165),(84,166),(85,167),(86,168),(87,113),(88,114),(89,115),(90,116),(91,117),(92,118),(93,119),(94,120),(95,121),(96,122),(97,123),(98,124),(99,125),(100,126),(101,127),(102,128),(103,129),(104,130),(105,131),(106,132),(107,133),(108,134),(109,135),(110,136),(111,137),(112,138)], [(1,191),(2,192),(3,193),(4,194),(5,195),(6,196),(7,197),(8,198),(9,199),(10,200),(11,201),(12,202),(13,203),(14,204),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,211),(22,212),(23,213),(24,214),(25,215),(26,216),(27,217),(28,218),(29,219),(30,220),(31,221),(32,222),(33,223),(34,224),(35,169),(36,170),(37,171),(38,172),(39,173),(40,174),(41,175),(42,176),(43,177),(44,178),(45,179),(46,180),(47,181),(48,182),(49,183),(50,184),(51,185),(52,186),(53,187),(54,188),(55,189),(56,190),(57,167),(58,168),(59,113),(60,114),(61,115),(62,116),(63,117),(64,118),(65,119),(66,120),(67,121),(68,122),(69,123),(70,124),(71,125),(72,126),(73,127),(74,128),(75,129),(76,130),(77,131),(78,132),(79,133),(80,134),(81,135),(82,136),(83,137),(84,138),(85,139),(86,140),(87,141),(88,142),(89,143),(90,144),(91,145),(92,146),(93,147),(94,148),(95,149),(96,150),(97,151),(98,152),(99,153),(100,154),(101,155),(102,156),(103,157),(104,158),(105,159),(106,160),(107,161),(108,162),(109,163),(110,164),(111,165),(112,166)], [(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(169,197),(170,198),(171,199),(172,200),(173,201),(174,202),(175,203),(176,204),(177,205),(178,206),(179,207),(180,208),(181,209),(182,210),(183,211),(184,212),(185,213),(186,214),(187,215),(188,216),(189,217),(190,218),(191,219),(192,220),(193,221),(194,222),(195,223),(196,224)], [(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)], [(1,70,219,152),(2,97,220,123),(3,68,221,150),(4,95,222,121),(5,66,223,148),(6,93,224,119),(7,64,169,146),(8,91,170,117),(9,62,171,144),(10,89,172,115),(11,60,173,142),(12,87,174,113),(13,58,175,140),(14,85,176,167),(15,112,177,138),(16,83,178,165),(17,110,179,136),(18,81,180,163),(19,108,181,134),(20,79,182,161),(21,106,183,132),(22,77,184,159),(23,104,185,130),(24,75,186,157),(25,102,187,128),(26,73,188,155),(27,100,189,126),(28,71,190,153),(29,98,191,124),(30,69,192,151),(31,96,193,122),(32,67,194,149),(33,94,195,120),(34,65,196,147),(35,92,197,118),(36,63,198,145),(37,90,199,116),(38,61,200,143),(39,88,201,114),(40,59,202,141),(41,86,203,168),(42,57,204,139),(43,84,205,166),(44,111,206,137),(45,82,207,164),(46,109,208,135),(47,80,209,162),(48,107,210,133),(49,78,211,160),(50,105,212,131),(51,76,213,158),(52,103,214,129),(53,74,215,156),(54,101,216,127),(55,72,217,154),(56,99,218,125)]])
124 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 7A | 7B | 7C | 8A | ··· | 8H | 14A | ··· | 14U | 28A | ··· | 28X | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 28 | ··· | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | - | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | Q8 | D4 | D7 | Dic7 | D14 | D14 | C4○D8 | Dic14 | D28 | D28 | D56⋊7C2 |
| kernel | C23.22D28 | C8⋊Dic7 | C56⋊1C4 | C23.21D14 | C22×C56 | C2×C56 | C2×C28 | C2×C28 | C22×C14 | C22×C8 | C2×C8 | C2×C8 | C22×C4 | C14 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 8 | 1 | 2 | 1 | 3 | 12 | 6 | 3 | 8 | 12 | 6 | 6 | 48 |
Matrix representation of C23.22D28 ►in GL3(𝔽113) generated by
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 112 |
| 112 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
| 0 | 112 | 0 |
| 0 | 0 | 112 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 26 |
| 98 | 0 | 0 |
| 0 | 0 | 26 |
| 0 | 13 | 0 |
G:=sub<GL(3,GF(113))| [1,0,0,0,1,0,0,0,112],[112,0,0,0,1,0,0,0,1],[1,0,0,0,112,0,0,0,112],[1,0,0,0,13,0,0,0,26],[98,0,0,0,0,13,0,26,0] >;
C23.22D28 in GAP, Magma, Sage, TeX
C_2^3._{22}D_{28} % in TeX
G:=Group("C2^3.22D28"); // GroupNames label
G:=SmallGroup(448,640);
// by ID
G=gap.SmallGroup(448,640);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,232,422,100,1684,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^28=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^27>;
// generators/relations