direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.21D14, C24.68D14, (C22×C28)⋊16C4, (C23×C4).12D7, (C23×C28).17C2, C14.43(C23×C4), C4⋊Dic7⋊83C22, (C22×C4)⋊10Dic7, C14⋊4(C42⋊C2), C2.5(C23×Dic7), C28.181(C22×C4), (C2×C28).885C23, (C2×C14).284C24, (C4×Dic7)⋊81C22, (C22×C4).448D14, C23.36(C2×Dic7), C4.39(C22×Dic7), C22.41(C23×D7), C22.80(C4○D28), C23.232(C22×D7), (C22×C14).413C23, (C22×C28).547C22, (C23×C14).106C22, (C2×Dic7).278C23, C23.D7.144C22, C22.31(C22×Dic7), (C22×Dic7).229C22, (C2×C28)⋊37(C2×C4), C7⋊5(C2×C42⋊C2), (C2×C4×Dic7)⋊37C2, C2.5(C2×C4○D28), (C2×C4⋊Dic7)⋊50C2, C14.60(C2×C4○D4), (C2×C4)⋊11(C2×Dic7), (C2×C4).829(C22×D7), (C2×C23.D7).25C2, (C2×C14).111(C4○D4), (C2×C14).208(C22×C4), (C22×C14).143(C2×C4), SmallGroup(448,1239)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.21D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e14=d, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef-1=e13 >
Subgroups: 900 in 330 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C2×Dic7, C2×Dic7, C2×C28, C22×C14, C22×C14, C22×C14, C2×C42⋊C2, C4×Dic7, C4⋊Dic7, C23.D7, C22×Dic7, C22×C28, C22×C28, C23×C14, C2×C4×Dic7, C2×C4⋊Dic7, C23.21D14, C2×C23.D7, C23×C28, C2×C23.21D14
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, C24, Dic7, D14, C42⋊C2, C23×C4, C2×C4○D4, C2×Dic7, C22×D7, C2×C42⋊C2, C4○D28, C22×Dic7, C23×D7, C23.21D14, C2×C4○D28, C23×Dic7, C2×C23.21D14
►On 224 points
Generators in S224
(1 160)(2 161)(3 162)(4 163)(5 164)(6 165)(7 166)(8 167)(9 168)(10 141)(11 142)(12 143)(13 144)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 151)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 106)(38 107)(39 108)(40 109)(41 110)(42 111)(43 112)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 97)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 113)(84 114)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 211)(179 212)(180 213)(181 214)(182 215)(183 216)(184 217)(185 218)(186 219)(187 220)(188 221)(189 222)(190 223)(191 224)(192 197)(193 198)(194 199)(195 200)(196 201)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)
(1 182)(2 183)(3 184)(4 185)(5 186)(6 187)(7 188)(8 189)(9 190)(10 191)(11 192)(12 193)(13 194)(14 195)(15 196)(16 169)(17 170)(18 171)(19 172)(20 173)(21 174)(22 175)(23 176)(24 177)(25 178)(26 179)(27 180)(28 181)(29 131)(30 132)(31 133)(32 134)(33 135)(34 136)(35 137)(36 138)(37 139)(38 140)(39 113)(40 114)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 121)(48 122)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 129)(56 130)(57 110)(58 111)(59 112)(60 85)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 97)(73 98)(74 99)(75 100)(76 101)(77 102)(78 103)(79 104)(80 105)(81 106)(82 107)(83 108)(84 109)(141 224)(142 197)(143 198)(144 199)(145 200)(146 201)(147 202)(148 203)(149 204)(150 205)(151 206)(152 207)(153 208)(154 209)(155 210)(156 211)(157 212)(158 213)(159 214)(160 215)(161 216)(162 217)(163 218)(164 219)(165 220)(166 221)(167 222)(168 223)
(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 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)
(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 70 196 109)(2 83 169 94)(3 68 170 107)(4 81 171 92)(5 66 172 105)(6 79 173 90)(7 64 174 103)(8 77 175 88)(9 62 176 101)(10 75 177 86)(11 60 178 99)(12 73 179 112)(13 58 180 97)(14 71 181 110)(15 84 182 95)(16 69 183 108)(17 82 184 93)(18 67 185 106)(19 80 186 91)(20 65 187 104)(21 78 188 89)(22 63 189 102)(23 76 190 87)(24 61 191 100)(25 74 192 85)(26 59 193 98)(27 72 194 111)(28 57 195 96)(29 157 117 198)(30 142 118 211)(31 155 119 224)(32 168 120 209)(33 153 121 222)(34 166 122 207)(35 151 123 220)(36 164 124 205)(37 149 125 218)(38 162 126 203)(39 147 127 216)(40 160 128 201)(41 145 129 214)(42 158 130 199)(43 143 131 212)(44 156 132 197)(45 141 133 210)(46 154 134 223)(47 167 135 208)(48 152 136 221)(49 165 137 206)(50 150 138 219)(51 163 139 204)(52 148 140 217)(53 161 113 202)(54 146 114 215)(55 159 115 200)(56 144 116 213)
G:=sub<Sym(224)| (1,160)(2,161)(3,162)(4,163)(5,164)(6,165)(7,166)(8,167)(9,168)(10,141)(11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,113)(84,114)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213)(181,214)(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,221)(189,222)(190,223)(191,224)(192,197)(193,198)(194,199)(195,200)(196,201), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140), (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,181)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,110)(58,111)(59,112)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(141,224)(142,197)(143,198)(144,199)(145,200)(146,201)(147,202)(148,203)(149,204)(150,205)(151,206)(152,207)(153,208)(154,209)(155,210)(156,211)(157,212)(158,213)(159,214)(160,215)(161,216)(162,217)(163,218)(164,219)(165,220)(166,221)(167,222)(168,223), (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)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (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), (1,70,196,109)(2,83,169,94)(3,68,170,107)(4,81,171,92)(5,66,172,105)(6,79,173,90)(7,64,174,103)(8,77,175,88)(9,62,176,101)(10,75,177,86)(11,60,178,99)(12,73,179,112)(13,58,180,97)(14,71,181,110)(15,84,182,95)(16,69,183,108)(17,82,184,93)(18,67,185,106)(19,80,186,91)(20,65,187,104)(21,78,188,89)(22,63,189,102)(23,76,190,87)(24,61,191,100)(25,74,192,85)(26,59,193,98)(27,72,194,111)(28,57,195,96)(29,157,117,198)(30,142,118,211)(31,155,119,224)(32,168,120,209)(33,153,121,222)(34,166,122,207)(35,151,123,220)(36,164,124,205)(37,149,125,218)(38,162,126,203)(39,147,127,216)(40,160,128,201)(41,145,129,214)(42,158,130,199)(43,143,131,212)(44,156,132,197)(45,141,133,210)(46,154,134,223)(47,167,135,208)(48,152,136,221)(49,165,137,206)(50,150,138,219)(51,163,139,204)(52,148,140,217)(53,161,113,202)(54,146,114,215)(55,159,115,200)(56,144,116,213)>;
G:=Group( (1,160)(2,161)(3,162)(4,163)(5,164)(6,165)(7,166)(8,167)(9,168)(10,141)(11,142)(12,143)(13,144)(14,145)(15,146)(16,147)(17,148)(18,149)(19,150)(20,151)(21,152)(22,153)(23,154)(24,155)(25,156)(26,157)(27,158)(28,159)(29,98)(30,99)(31,100)(32,101)(33,102)(34,103)(35,104)(36,105)(37,106)(38,107)(39,108)(40,109)(41,110)(42,111)(43,112)(44,85)(45,86)(46,87)(47,88)(48,89)(49,90)(50,91)(51,92)(52,93)(53,94)(54,95)(55,96)(56,97)(57,115)(58,116)(59,117)(60,118)(61,119)(62,120)(63,121)(64,122)(65,123)(66,124)(67,125)(68,126)(69,127)(70,128)(71,129)(72,130)(73,131)(74,132)(75,133)(76,134)(77,135)(78,136)(79,137)(80,138)(81,139)(82,140)(83,113)(84,114)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213)(181,214)(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,221)(189,222)(190,223)(191,224)(192,197)(193,198)(194,199)(195,200)(196,201), (29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140), (1,182)(2,183)(3,184)(4,185)(5,186)(6,187)(7,188)(8,189)(9,190)(10,191)(11,192)(12,193)(13,194)(14,195)(15,196)(16,169)(17,170)(18,171)(19,172)(20,173)(21,174)(22,175)(23,176)(24,177)(25,178)(26,179)(27,180)(28,181)(29,131)(30,132)(31,133)(32,134)(33,135)(34,136)(35,137)(36,138)(37,139)(38,140)(39,113)(40,114)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,121)(48,122)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,129)(56,130)(57,110)(58,111)(59,112)(60,85)(61,86)(62,87)(63,88)(64,89)(65,90)(66,91)(67,92)(68,93)(69,94)(70,95)(71,96)(72,97)(73,98)(74,99)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(141,224)(142,197)(143,198)(144,199)(145,200)(146,201)(147,202)(148,203)(149,204)(150,205)(151,206)(152,207)(153,208)(154,209)(155,210)(156,211)(157,212)(158,213)(159,214)(160,215)(161,216)(162,217)(163,218)(164,219)(165,220)(166,221)(167,222)(168,223), (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)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112)(113,127)(114,128)(115,129)(116,130)(117,131)(118,132)(119,133)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140)(141,155)(142,156)(143,157)(144,158)(145,159)(146,160)(147,161)(148,162)(149,163)(150,164)(151,165)(152,166)(153,167)(154,168)(169,183)(170,184)(171,185)(172,186)(173,187)(174,188)(175,189)(176,190)(177,191)(178,192)(179,193)(180,194)(181,195)(182,196)(197,211)(198,212)(199,213)(200,214)(201,215)(202,216)(203,217)(204,218)(205,219)(206,220)(207,221)(208,222)(209,223)(210,224), (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), (1,70,196,109)(2,83,169,94)(3,68,170,107)(4,81,171,92)(5,66,172,105)(6,79,173,90)(7,64,174,103)(8,77,175,88)(9,62,176,101)(10,75,177,86)(11,60,178,99)(12,73,179,112)(13,58,180,97)(14,71,181,110)(15,84,182,95)(16,69,183,108)(17,82,184,93)(18,67,185,106)(19,80,186,91)(20,65,187,104)(21,78,188,89)(22,63,189,102)(23,76,190,87)(24,61,191,100)(25,74,192,85)(26,59,193,98)(27,72,194,111)(28,57,195,96)(29,157,117,198)(30,142,118,211)(31,155,119,224)(32,168,120,209)(33,153,121,222)(34,166,122,207)(35,151,123,220)(36,164,124,205)(37,149,125,218)(38,162,126,203)(39,147,127,216)(40,160,128,201)(41,145,129,214)(42,158,130,199)(43,143,131,212)(44,156,132,197)(45,141,133,210)(46,154,134,223)(47,167,135,208)(48,152,136,221)(49,165,137,206)(50,150,138,219)(51,163,139,204)(52,148,140,217)(53,161,113,202)(54,146,114,215)(55,159,115,200)(56,144,116,213) );
G=PermutationGroup([[(1,160),(2,161),(3,162),(4,163),(5,164),(6,165),(7,166),(8,167),(9,168),(10,141),(11,142),(12,143),(13,144),(14,145),(15,146),(16,147),(17,148),(18,149),(19,150),(20,151),(21,152),(22,153),(23,154),(24,155),(25,156),(26,157),(27,158),(28,159),(29,98),(30,99),(31,100),(32,101),(33,102),(34,103),(35,104),(36,105),(37,106),(38,107),(39,108),(40,109),(41,110),(42,111),(43,112),(44,85),(45,86),(46,87),(47,88),(48,89),(49,90),(50,91),(51,92),(52,93),(53,94),(54,95),(55,96),(56,97),(57,115),(58,116),(59,117),(60,118),(61,119),(62,120),(63,121),(64,122),(65,123),(66,124),(67,125),(68,126),(69,127),(70,128),(71,129),(72,130),(73,131),(74,132),(75,133),(76,134),(77,135),(78,136),(79,137),(80,138),(81,139),(82,140),(83,113),(84,114),(169,202),(170,203),(171,204),(172,205),(173,206),(174,207),(175,208),(176,209),(177,210),(178,211),(179,212),(180,213),(181,214),(182,215),(183,216),(184,217),(185,218),(186,219),(187,220),(188,221),(189,222),(190,223),(191,224),(192,197),(193,198),(194,199),(195,200),(196,201)], [(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140)], [(1,182),(2,183),(3,184),(4,185),(5,186),(6,187),(7,188),(8,189),(9,190),(10,191),(11,192),(12,193),(13,194),(14,195),(15,196),(16,169),(17,170),(18,171),(19,172),(20,173),(21,174),(22,175),(23,176),(24,177),(25,178),(26,179),(27,180),(28,181),(29,131),(30,132),(31,133),(32,134),(33,135),(34,136),(35,137),(36,138),(37,139),(38,140),(39,113),(40,114),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,121),(48,122),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,129),(56,130),(57,110),(58,111),(59,112),(60,85),(61,86),(62,87),(63,88),(64,89),(65,90),(66,91),(67,92),(68,93),(69,94),(70,95),(71,96),(72,97),(73,98),(74,99),(75,100),(76,101),(77,102),(78,103),(79,104),(80,105),(81,106),(82,107),(83,108),(84,109),(141,224),(142,197),(143,198),(144,199),(145,200),(146,201),(147,202),(148,203),(149,204),(150,205),(151,206),(152,207),(153,208),(154,209),(155,210),(156,211),(157,212),(158,213),(159,214),(160,215),(161,216),(162,217),(163,218),(164,219),(165,220),(166,221),(167,222),(168,223)], [(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),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112),(113,127),(114,128),(115,129),(116,130),(117,131),(118,132),(119,133),(120,134),(121,135),(122,136),(123,137),(124,138),(125,139),(126,140),(141,155),(142,156),(143,157),(144,158),(145,159),(146,160),(147,161),(148,162),(149,163),(150,164),(151,165),(152,166),(153,167),(154,168),(169,183),(170,184),(171,185),(172,186),(173,187),(174,188),(175,189),(176,190),(177,191),(178,192),(179,193),(180,194),(181,195),(182,196),(197,211),(198,212),(199,213),(200,214),(201,215),(202,216),(203,217),(204,218),(205,219),(206,220),(207,221),(208,222),(209,223),(210,224)], [(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)], [(1,70,196,109),(2,83,169,94),(3,68,170,107),(4,81,171,92),(5,66,172,105),(6,79,173,90),(7,64,174,103),(8,77,175,88),(9,62,176,101),(10,75,177,86),(11,60,178,99),(12,73,179,112),(13,58,180,97),(14,71,181,110),(15,84,182,95),(16,69,183,108),(17,82,184,93),(18,67,185,106),(19,80,186,91),(20,65,187,104),(21,78,188,89),(22,63,189,102),(23,76,190,87),(24,61,191,100),(25,74,192,85),(26,59,193,98),(27,72,194,111),(28,57,195,96),(29,157,117,198),(30,142,118,211),(31,155,119,224),(32,168,120,209),(33,153,121,222),(34,166,122,207),(35,151,123,220),(36,164,124,205),(37,149,125,218),(38,162,126,203),(39,147,127,216),(40,160,128,201),(41,145,129,214),(42,158,130,199),(43,143,131,212),(44,156,132,197),(45,141,133,210),(46,154,134,223),(47,167,135,208),(48,152,136,221),(49,165,137,206),(50,150,138,219),(51,163,139,204),(52,148,140,217),(53,161,113,202),(54,146,114,215),(55,159,115,200),(56,144,116,213)]])
136 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 4M | ··· | 4AB | 7A | 7B | 7C | 14A | ··· | 14AS | 28A | ··· | 28AV |
| order | 1 | 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 | 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 | 2 |
| type | + | + | + | + | + | + | + | - | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | Dic7 | D14 | D14 | C4○D28 |
| kernel | C2×C23.21D14 | C2×C4×Dic7 | C2×C4⋊Dic7 | C23.21D14 | C2×C23.D7 | C23×C28 | C22×C28 | C23×C4 | C2×C14 | C22×C4 | C22×C4 | C24 | C22 |
| # reps | 1 | 2 | 2 | 8 | 2 | 1 | 16 | 3 | 8 | 24 | 18 | 3 | 48 |
Matrix representation of C2×C23.21D14 ►in GL5(𝔽29)
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 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 | 14 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 |
G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,14,0,0,0,0,0,2],[17,0,0,0,0,0,0,17,0,0,0,12,0,0,0,0,0,0,0,28,0,0,0,1,0] >;
C2×C23.21D14 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{21}D_{14} % in TeX
G:=Group("C2xC2^3.21D14"); // GroupNames label
G:=SmallGroup(448,1239);
// by ID
G=gap.SmallGroup(448,1239);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,184,1123,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^14=d,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^13>;
// generators/relations