metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.45D28, C24.49D14, C14.44(C4×D4), (C2×Dic7)⋊17D4, C23.26(C4×D7), C22⋊1(D14⋊C4), C14.32C22≀C2, (C23×Dic7)⋊1C2, (C22×C4).33D14, C22.44(C2×D28), (C22×C14).68D4, C22.103(D4×D7), C2.5(C22⋊D28), C14.86(C4⋊D4), C7⋊3(C23.23D4), C23.36(C7⋊D4), C14.C42⋊16C2, C2.4(Dic7⋊D4), (C22×C28).26C22, (C23×C14).41C22, (C23×D7).14C22, C23.285(C22×D7), C2.29(Dic7⋊4D4), C22.50(D4⋊2D7), (C22×C14).332C23, C2.5(C22.D28), C14.33(C22.D4), (C22×Dic7).186C22, (C2×C7⋊D4)⋊6C4, (C2×D14⋊C4)⋊6C2, (C2×C22⋊C4)⋊5D7, (C14×C22⋊C4)⋊3C2, C2.10(C2×D14⋊C4), (C2×Dic7)⋊6(C2×C4), (C22×D7)⋊4(C2×C4), C22.129(C2×C4×D7), (C2×C14)⋊2(C22⋊C4), (C2×C14).324(C2×D4), C14.37(C2×C22⋊C4), (C22×C7⋊D4).4C2, C22.53(C2×C7⋊D4), (C22×C14).55(C2×C4), (C2×C14).147(C4○D4), (C2×C14).111(C22×C4), SmallGroup(448,492)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.45D28
G = < a,b,c,d,e | a2=b2=c2=d28=1, e2=b, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 1428 in 286 conjugacy classes, 83 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, Dic7, C28, D14, C2×C14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C22×D4, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C23.23D4, D14⋊C4, C7×C22⋊C4, C22×Dic7, C22×Dic7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, C23×D7, C23×C14, C14.C42, C2×D14⋊C4, C14×C22⋊C4, C23×Dic7, C22×C7⋊D4, C23.45D28
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, C4○D4, D14, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22.D4, C4×D7, D28, C7⋊D4, C22×D7, C23.23D4, D14⋊C4, C2×C4×D7, C2×D28, D4×D7, D4⋊2D7, C2×C7⋊D4, Dic7⋊4D4, C22⋊D28, C22.D28, C2×D14⋊C4, Dic7⋊D4, C23.45D28
►On 224 points
Generators in S224
(1 101)(2 133)(3 103)(4 135)(5 105)(6 137)(7 107)(8 139)(9 109)(10 113)(11 111)(12 115)(13 85)(14 117)(15 87)(16 119)(17 89)(18 121)(19 91)(20 123)(21 93)(22 125)(23 95)(24 127)(25 97)(26 129)(27 99)(28 131)(29 210)(30 188)(31 212)(32 190)(33 214)(34 192)(35 216)(36 194)(37 218)(38 196)(39 220)(40 170)(41 222)(42 172)(43 224)(44 174)(45 198)(46 176)(47 200)(48 178)(49 202)(50 180)(51 204)(52 182)(53 206)(54 184)(55 208)(56 186)(57 187)(58 211)(59 189)(60 213)(61 191)(62 215)(63 193)(64 217)(65 195)(66 219)(67 169)(68 221)(69 171)(70 223)(71 173)(72 197)(73 175)(74 199)(75 177)(76 201)(77 179)(78 203)(79 181)(80 205)(81 183)(82 207)(83 185)(84 209)(86 164)(88 166)(90 168)(92 142)(94 144)(96 146)(98 148)(100 150)(102 152)(104 154)(106 156)(108 158)(110 160)(112 162)(114 161)(116 163)(118 165)(120 167)(122 141)(124 143)(126 145)(128 147)(130 149)(132 151)(134 153)(136 155)(138 157)(140 159)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 210)(30 211)(31 212)(32 213)(33 214)(34 215)(35 216)(36 217)(37 218)(38 219)(39 220)(40 221)(41 222)(42 223)(43 224)(44 197)(45 198)(46 199)(47 200)(48 201)(49 202)(50 203)(51 204)(52 205)(53 206)(54 207)(55 208)(56 209)(57 187)(58 188)(59 189)(60 190)(61 191)(62 192)(63 193)(64 194)(65 195)(66 196)(67 169)(68 170)(69 171)(70 172)(71 173)(72 174)(73 175)(74 176)(75 177)(76 178)(77 179)(78 180)(79 181)(80 182)(81 183)(82 184)(83 185)(84 186)(113 160)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)(121 168)(122 141)(123 142)(124 143)(125 144)(126 145)(127 146)(128 147)(129 148)(130 149)(131 150)(132 151)(133 152)(134 153)(135 154)(136 155)(137 156)(138 157)(139 158)(140 159)
(1 151)(2 152)(3 153)(4 154)(5 155)(6 156)(7 157)(8 158)(9 159)(10 160)(11 161)(12 162)(13 163)(14 164)(15 165)(16 166)(17 167)(18 168)(19 141)(20 142)(21 143)(22 144)(23 145)(24 146)(25 147)(26 148)(27 149)(28 150)(29 57)(30 58)(31 59)(32 60)(33 61)(34 62)(35 63)(36 64)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(85 116)(86 117)(87 118)(88 119)(89 120)(90 121)(91 122)(92 123)(93 124)(94 125)(95 126)(96 127)(97 128)(98 129)(99 130)(100 131)(101 132)(102 133)(103 134)(104 135)(105 136)(106 137)(107 138)(108 139)(109 140)(110 113)(111 114)(112 115)(169 220)(170 221)(171 222)(172 223)(173 224)(174 197)(175 198)(176 199)(177 200)(178 201)(179 202)(180 203)(181 204)(182 205)(183 206)(184 207)(185 208)(186 209)(187 210)(188 211)(189 212)(190 213)(191 214)(192 215)(193 216)(194 217)(195 218)(196 219)
(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)
(1 193 101 63)(2 62 102 192)(3 191 103 61)(4 60 104 190)(5 189 105 59)(6 58 106 188)(7 187 107 57)(8 84 108 186)(9 185 109 83)(10 82 110 184)(11 183 111 81)(12 80 112 182)(13 181 85 79)(14 78 86 180)(15 179 87 77)(16 76 88 178)(17 177 89 75)(18 74 90 176)(19 175 91 73)(20 72 92 174)(21 173 93 71)(22 70 94 172)(23 171 95 69)(24 68 96 170)(25 169 97 67)(26 66 98 196)(27 195 99 65)(28 64 100 194)(29 157 210 138)(30 137 211 156)(31 155 212 136)(32 135 213 154)(33 153 214 134)(34 133 215 152)(35 151 216 132)(36 131 217 150)(37 149 218 130)(38 129 219 148)(39 147 220 128)(40 127 221 146)(41 145 222 126)(42 125 223 144)(43 143 224 124)(44 123 197 142)(45 141 198 122)(46 121 199 168)(47 167 200 120)(48 119 201 166)(49 165 202 118)(50 117 203 164)(51 163 204 116)(52 115 205 162)(53 161 206 114)(54 113 207 160)(55 159 208 140)(56 139 209 158)
G:=sub<Sym(224)| (1,101)(2,133)(3,103)(4,135)(5,105)(6,137)(7,107)(8,139)(9,109)(10,113)(11,111)(12,115)(13,85)(14,117)(15,87)(16,119)(17,89)(18,121)(19,91)(20,123)(21,93)(22,125)(23,95)(24,127)(25,97)(26,129)(27,99)(28,131)(29,210)(30,188)(31,212)(32,190)(33,214)(34,192)(35,216)(36,194)(37,218)(38,196)(39,220)(40,170)(41,222)(42,172)(43,224)(44,174)(45,198)(46,176)(47,200)(48,178)(49,202)(50,180)(51,204)(52,182)(53,206)(54,184)(55,208)(56,186)(57,187)(58,211)(59,189)(60,213)(61,191)(62,215)(63,193)(64,217)(65,195)(66,219)(67,169)(68,221)(69,171)(70,223)(71,173)(72,197)(73,175)(74,199)(75,177)(76,201)(77,179)(78,203)(79,181)(80,205)(81,183)(82,207)(83,185)(84,209)(86,164)(88,166)(90,168)(92,142)(94,144)(96,146)(98,148)(100,150)(102,152)(104,154)(106,156)(108,158)(110,160)(112,162)(114,161)(116,163)(118,165)(120,167)(122,141)(124,143)(126,145)(128,147)(130,149)(132,151)(134,153)(136,155)(138,157)(140,159), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,210)(30,211)(31,212)(32,213)(33,214)(34,215)(35,216)(36,217)(37,218)(38,219)(39,220)(40,221)(41,222)(42,223)(43,224)(44,197)(45,198)(46,199)(47,200)(48,201)(49,202)(50,203)(51,204)(52,205)(53,206)(54,207)(55,208)(56,209)(57,187)(58,188)(59,189)(60,190)(61,191)(62,192)(63,193)(64,194)(65,195)(66,196)(67,169)(68,170)(69,171)(70,172)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,181)(80,182)(81,183)(82,184)(83,185)(84,186)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152)(134,153)(135,154)(136,155)(137,156)(138,157)(139,158)(140,159), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,161)(12,162)(13,163)(14,164)(15,165)(16,166)(17,167)(18,168)(19,141)(20,142)(21,143)(22,144)(23,145)(24,146)(25,147)(26,148)(27,149)(28,150)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,113)(111,114)(112,115)(169,220)(170,221)(171,222)(172,223)(173,224)(174,197)(175,198)(176,199)(177,200)(178,201)(179,202)(180,203)(181,204)(182,205)(183,206)(184,207)(185,208)(186,209)(187,210)(188,211)(189,212)(190,213)(191,214)(192,215)(193,216)(194,217)(195,218)(196,219), (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), (1,193,101,63)(2,62,102,192)(3,191,103,61)(4,60,104,190)(5,189,105,59)(6,58,106,188)(7,187,107,57)(8,84,108,186)(9,185,109,83)(10,82,110,184)(11,183,111,81)(12,80,112,182)(13,181,85,79)(14,78,86,180)(15,179,87,77)(16,76,88,178)(17,177,89,75)(18,74,90,176)(19,175,91,73)(20,72,92,174)(21,173,93,71)(22,70,94,172)(23,171,95,69)(24,68,96,170)(25,169,97,67)(26,66,98,196)(27,195,99,65)(28,64,100,194)(29,157,210,138)(30,137,211,156)(31,155,212,136)(32,135,213,154)(33,153,214,134)(34,133,215,152)(35,151,216,132)(36,131,217,150)(37,149,218,130)(38,129,219,148)(39,147,220,128)(40,127,221,146)(41,145,222,126)(42,125,223,144)(43,143,224,124)(44,123,197,142)(45,141,198,122)(46,121,199,168)(47,167,200,120)(48,119,201,166)(49,165,202,118)(50,117,203,164)(51,163,204,116)(52,115,205,162)(53,161,206,114)(54,113,207,160)(55,159,208,140)(56,139,209,158)>;
G:=Group( (1,101)(2,133)(3,103)(4,135)(5,105)(6,137)(7,107)(8,139)(9,109)(10,113)(11,111)(12,115)(13,85)(14,117)(15,87)(16,119)(17,89)(18,121)(19,91)(20,123)(21,93)(22,125)(23,95)(24,127)(25,97)(26,129)(27,99)(28,131)(29,210)(30,188)(31,212)(32,190)(33,214)(34,192)(35,216)(36,194)(37,218)(38,196)(39,220)(40,170)(41,222)(42,172)(43,224)(44,174)(45,198)(46,176)(47,200)(48,178)(49,202)(50,180)(51,204)(52,182)(53,206)(54,184)(55,208)(56,186)(57,187)(58,211)(59,189)(60,213)(61,191)(62,215)(63,193)(64,217)(65,195)(66,219)(67,169)(68,221)(69,171)(70,223)(71,173)(72,197)(73,175)(74,199)(75,177)(76,201)(77,179)(78,203)(79,181)(80,205)(81,183)(82,207)(83,185)(84,209)(86,164)(88,166)(90,168)(92,142)(94,144)(96,146)(98,148)(100,150)(102,152)(104,154)(106,156)(108,158)(110,160)(112,162)(114,161)(116,163)(118,165)(120,167)(122,141)(124,143)(126,145)(128,147)(130,149)(132,151)(134,153)(136,155)(138,157)(140,159), (1,101)(2,102)(3,103)(4,104)(5,105)(6,106)(7,107)(8,108)(9,109)(10,110)(11,111)(12,112)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,97)(26,98)(27,99)(28,100)(29,210)(30,211)(31,212)(32,213)(33,214)(34,215)(35,216)(36,217)(37,218)(38,219)(39,220)(40,221)(41,222)(42,223)(43,224)(44,197)(45,198)(46,199)(47,200)(48,201)(49,202)(50,203)(51,204)(52,205)(53,206)(54,207)(55,208)(56,209)(57,187)(58,188)(59,189)(60,190)(61,191)(62,192)(63,193)(64,194)(65,195)(66,196)(67,169)(68,170)(69,171)(70,172)(71,173)(72,174)(73,175)(74,176)(75,177)(76,178)(77,179)(78,180)(79,181)(80,182)(81,183)(82,184)(83,185)(84,186)(113,160)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167)(121,168)(122,141)(123,142)(124,143)(125,144)(126,145)(127,146)(128,147)(129,148)(130,149)(131,150)(132,151)(133,152)(134,153)(135,154)(136,155)(137,156)(138,157)(139,158)(140,159), (1,151)(2,152)(3,153)(4,154)(5,155)(6,156)(7,157)(8,158)(9,159)(10,160)(11,161)(12,162)(13,163)(14,164)(15,165)(16,166)(17,167)(18,168)(19,141)(20,142)(21,143)(22,144)(23,145)(24,146)(25,147)(26,148)(27,149)(28,150)(29,57)(30,58)(31,59)(32,60)(33,61)(34,62)(35,63)(36,64)(37,65)(38,66)(39,67)(40,68)(41,69)(42,70)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(85,116)(86,117)(87,118)(88,119)(89,120)(90,121)(91,122)(92,123)(93,124)(94,125)(95,126)(96,127)(97,128)(98,129)(99,130)(100,131)(101,132)(102,133)(103,134)(104,135)(105,136)(106,137)(107,138)(108,139)(109,140)(110,113)(111,114)(112,115)(169,220)(170,221)(171,222)(172,223)(173,224)(174,197)(175,198)(176,199)(177,200)(178,201)(179,202)(180,203)(181,204)(182,205)(183,206)(184,207)(185,208)(186,209)(187,210)(188,211)(189,212)(190,213)(191,214)(192,215)(193,216)(194,217)(195,218)(196,219), (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), (1,193,101,63)(2,62,102,192)(3,191,103,61)(4,60,104,190)(5,189,105,59)(6,58,106,188)(7,187,107,57)(8,84,108,186)(9,185,109,83)(10,82,110,184)(11,183,111,81)(12,80,112,182)(13,181,85,79)(14,78,86,180)(15,179,87,77)(16,76,88,178)(17,177,89,75)(18,74,90,176)(19,175,91,73)(20,72,92,174)(21,173,93,71)(22,70,94,172)(23,171,95,69)(24,68,96,170)(25,169,97,67)(26,66,98,196)(27,195,99,65)(28,64,100,194)(29,157,210,138)(30,137,211,156)(31,155,212,136)(32,135,213,154)(33,153,214,134)(34,133,215,152)(35,151,216,132)(36,131,217,150)(37,149,218,130)(38,129,219,148)(39,147,220,128)(40,127,221,146)(41,145,222,126)(42,125,223,144)(43,143,224,124)(44,123,197,142)(45,141,198,122)(46,121,199,168)(47,167,200,120)(48,119,201,166)(49,165,202,118)(50,117,203,164)(51,163,204,116)(52,115,205,162)(53,161,206,114)(54,113,207,160)(55,159,208,140)(56,139,209,158) );
G=PermutationGroup([[(1,101),(2,133),(3,103),(4,135),(5,105),(6,137),(7,107),(8,139),(9,109),(10,113),(11,111),(12,115),(13,85),(14,117),(15,87),(16,119),(17,89),(18,121),(19,91),(20,123),(21,93),(22,125),(23,95),(24,127),(25,97),(26,129),(27,99),(28,131),(29,210),(30,188),(31,212),(32,190),(33,214),(34,192),(35,216),(36,194),(37,218),(38,196),(39,220),(40,170),(41,222),(42,172),(43,224),(44,174),(45,198),(46,176),(47,200),(48,178),(49,202),(50,180),(51,204),(52,182),(53,206),(54,184),(55,208),(56,186),(57,187),(58,211),(59,189),(60,213),(61,191),(62,215),(63,193),(64,217),(65,195),(66,219),(67,169),(68,221),(69,171),(70,223),(71,173),(72,197),(73,175),(74,199),(75,177),(76,201),(77,179),(78,203),(79,181),(80,205),(81,183),(82,207),(83,185),(84,209),(86,164),(88,166),(90,168),(92,142),(94,144),(96,146),(98,148),(100,150),(102,152),(104,154),(106,156),(108,158),(110,160),(112,162),(114,161),(116,163),(118,165),(120,167),(122,141),(124,143),(126,145),(128,147),(130,149),(132,151),(134,153),(136,155),(138,157),(140,159)], [(1,101),(2,102),(3,103),(4,104),(5,105),(6,106),(7,107),(8,108),(9,109),(10,110),(11,111),(12,112),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,97),(26,98),(27,99),(28,100),(29,210),(30,211),(31,212),(32,213),(33,214),(34,215),(35,216),(36,217),(37,218),(38,219),(39,220),(40,221),(41,222),(42,223),(43,224),(44,197),(45,198),(46,199),(47,200),(48,201),(49,202),(50,203),(51,204),(52,205),(53,206),(54,207),(55,208),(56,209),(57,187),(58,188),(59,189),(60,190),(61,191),(62,192),(63,193),(64,194),(65,195),(66,196),(67,169),(68,170),(69,171),(70,172),(71,173),(72,174),(73,175),(74,176),(75,177),(76,178),(77,179),(78,180),(79,181),(80,182),(81,183),(82,184),(83,185),(84,186),(113,160),(114,161),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167),(121,168),(122,141),(123,142),(124,143),(125,144),(126,145),(127,146),(128,147),(129,148),(130,149),(131,150),(132,151),(133,152),(134,153),(135,154),(136,155),(137,156),(138,157),(139,158),(140,159)], [(1,151),(2,152),(3,153),(4,154),(5,155),(6,156),(7,157),(8,158),(9,159),(10,160),(11,161),(12,162),(13,163),(14,164),(15,165),(16,166),(17,167),(18,168),(19,141),(20,142),(21,143),(22,144),(23,145),(24,146),(25,147),(26,148),(27,149),(28,150),(29,57),(30,58),(31,59),(32,60),(33,61),(34,62),(35,63),(36,64),(37,65),(38,66),(39,67),(40,68),(41,69),(42,70),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(85,116),(86,117),(87,118),(88,119),(89,120),(90,121),(91,122),(92,123),(93,124),(94,125),(95,126),(96,127),(97,128),(98,129),(99,130),(100,131),(101,132),(102,133),(103,134),(104,135),(105,136),(106,137),(107,138),(108,139),(109,140),(110,113),(111,114),(112,115),(169,220),(170,221),(171,222),(172,223),(173,224),(174,197),(175,198),(176,199),(177,200),(178,201),(179,202),(180,203),(181,204),(182,205),(183,206),(184,207),(185,208),(186,209),(187,210),(188,211),(189,212),(190,213),(191,214),(192,215),(193,216),(194,217),(195,218),(196,219)], [(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)], [(1,193,101,63),(2,62,102,192),(3,191,103,61),(4,60,104,190),(5,189,105,59),(6,58,106,188),(7,187,107,57),(8,84,108,186),(9,185,109,83),(10,82,110,184),(11,183,111,81),(12,80,112,182),(13,181,85,79),(14,78,86,180),(15,179,87,77),(16,76,88,178),(17,177,89,75),(18,74,90,176),(19,175,91,73),(20,72,92,174),(21,173,93,71),(22,70,94,172),(23,171,95,69),(24,68,96,170),(25,169,97,67),(26,66,98,196),(27,195,99,65),(28,64,100,194),(29,157,210,138),(30,137,211,156),(31,155,212,136),(32,135,213,154),(33,153,214,134),(34,133,215,152),(35,151,216,132),(36,131,217,150),(37,149,218,130),(38,129,219,148),(39,147,220,128),(40,127,221,146),(41,145,222,126),(42,125,223,144),(43,143,224,124),(44,123,197,142),(45,141,198,122),(46,121,199,168),(47,167,200,120),(48,119,201,166),(49,165,202,118),(50,117,203,164),(51,163,204,116),(52,115,205,162),(53,161,206,114),(54,113,207,160),(55,159,208,140),(56,139,209,158)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AG | 28A | ··· | 28X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 28 | 28 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D7 | C4○D4 | D14 | D14 | C4×D7 | D28 | C7⋊D4 | D4×D7 | D4⋊2D7 |
| kernel | C23.45D28 | C14.C42 | C2×D14⋊C4 | C14×C22⋊C4 | C23×Dic7 | C22×C7⋊D4 | C2×C7⋊D4 | C2×Dic7 | C22×C14 | C2×C22⋊C4 | C2×C14 | C22×C4 | C24 | C23 | C23 | C23 | C22 | C22 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 4 | 4 | 3 | 4 | 6 | 3 | 12 | 12 | 12 | 6 | 6 |
Matrix representation of C23.45D28 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 17 | 16 | 0 | 0 | 0 | 0 |
| 13 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 25 | 0 | 0 |
| 0 | 0 | 4 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 23 |
| 0 | 0 | 0 | 0 | 18 | 14 |
| 22 | 22 | 0 | 0 | 0 | 0 |
| 3 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 19 | 0 | 0 |
| 0 | 0 | 28 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,24,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[17,13,0,0,0,0,16,19,0,0,0,0,0,0,20,4,0,0,0,0,25,21,0,0,0,0,0,0,15,18,0,0,0,0,23,14],[22,3,0,0,0,0,22,7,0,0,0,0,0,0,22,28,0,0,0,0,19,7,0,0,0,0,0,0,28,0,0,0,0,0,0,28] >;
C23.45D28 in GAP, Magma, Sage, TeX
C_2^3._{45}D_{28} % in TeX
G:=Group("C2^3.45D28"); // GroupNames label
G:=SmallGroup(448,492);
// by ID
G=gap.SmallGroup(448,492);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,477,422,387,58,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=1,e^2=b,a*b=b*a,d*a*d^-1=a*c=c*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^-1=b*d^-1>;
// generators/relations