direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.18D14, C24.58D14, (C2×D4).228D14, (C23×Dic7)⋊8C2, (C22×D4).10D7, (C2×C14).291C24, (C2×C28).642C23, Dic7⋊C4⋊72C22, (C22×C4).269D14, (C22×C14).121D4, C14.139(C22×D4), C23.67(C7⋊D4), C23.D7⋊57C22, (D4×C14).311C22, C14⋊5(C22.D4), (C23×C14).73C22, C23.133(C22×D7), C22.305(C23×D7), C22.77(D4⋊2D7), (C22×C28).437C22, (C22×C14).227C23, (C2×Dic7).281C23, (C22×Dic7)⋊48C22, (D4×C2×C14).21C2, (C2×C14).73(C2×D4), C7⋊6(C2×C22.D4), (C2×Dic7⋊C4)⋊47C2, C14.103(C2×C4○D4), C2.67(C2×D4⋊2D7), (C2×C23.D7)⋊24C2, C2.12(C22×C7⋊D4), (C2×C4).236(C22×D7), C22.108(C2×C7⋊D4), (C2×C14).175(C4○D4), SmallGroup(448,1249)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C23.18D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 1172 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C22.D4, Dic7⋊C4, C23.D7, C22×Dic7, C22×Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C2×Dic7⋊C4, C23.18D14, C2×C23.D7, C2×C23.D7, C23×Dic7, D4×C2×C14, C2×C23.18D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22.D4, C22×D4, C2×C4○D4, C7⋊D4, C22×D7, C2×C22.D4, D4⋊2D7, C2×C7⋊D4, C23×D7, C23.18D14, C2×D4⋊2D7, C22×C7⋊D4, C2×C23.18D14
►On 224 points
Generators in S224
(1 153)(2 154)(3 141)(4 142)(5 143)(6 144)(7 145)(8 146)(9 147)(10 148)(11 149)(12 150)(13 151)(14 152)(15 81)(16 82)(17 83)(18 84)(19 71)(20 72)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 201)(30 202)(31 203)(32 204)(33 205)(34 206)(35 207)(36 208)(37 209)(38 210)(39 197)(40 198)(41 199)(42 200)(43 86)(44 87)(45 88)(46 89)(47 90)(48 91)(49 92)(50 93)(51 94)(52 95)(53 96)(54 97)(55 98)(56 85)(57 172)(58 173)(59 174)(60 175)(61 176)(62 177)(63 178)(64 179)(65 180)(66 181)(67 182)(68 169)(69 170)(70 171)(99 162)(100 163)(101 164)(102 165)(103 166)(104 167)(105 168)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 161)(113 136)(114 137)(115 138)(116 139)(117 140)(118 127)(119 128)(120 129)(121 130)(122 131)(123 132)(124 133)(125 134)(126 135)(183 212)(184 213)(185 214)(186 215)(187 216)(188 217)(189 218)(190 219)(191 220)(192 221)(193 222)(194 223)(195 224)(196 211)
(1 25)(2 198)(3 27)(4 200)(5 15)(6 202)(7 17)(8 204)(9 19)(10 206)(11 21)(12 208)(13 23)(14 210)(16 183)(18 185)(20 187)(22 189)(24 191)(26 193)(28 195)(29 211)(30 144)(31 213)(32 146)(33 215)(34 148)(35 217)(36 150)(37 219)(38 152)(39 221)(40 154)(41 223)(42 142)(43 159)(44 131)(45 161)(46 133)(47 163)(48 135)(49 165)(50 137)(51 167)(52 139)(53 155)(54 127)(55 157)(56 129)(57 113)(58 103)(59 115)(60 105)(61 117)(62 107)(63 119)(64 109)(65 121)(66 111)(67 123)(68 99)(69 125)(70 101)(71 147)(72 216)(73 149)(74 218)(75 151)(76 220)(77 153)(78 222)(79 141)(80 224)(81 143)(82 212)(83 145)(84 214)(85 120)(86 110)(87 122)(88 112)(89 124)(90 100)(91 126)(92 102)(93 114)(94 104)(95 116)(96 106)(97 118)(98 108)(128 178)(130 180)(132 182)(134 170)(136 172)(138 174)(140 176)(156 177)(158 179)(160 181)(162 169)(164 171)(166 173)(168 175)(184 203)(186 205)(188 207)(190 209)(192 197)(194 199)(196 201)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 143)(16 144)(17 145)(18 146)(19 147)(20 148)(21 149)(22 150)(23 151)(24 152)(25 153)(26 154)(27 141)(28 142)(29 196)(30 183)(31 184)(32 185)(33 186)(34 187)(35 188)(36 189)(37 190)(38 191)(39 192)(40 193)(41 194)(42 195)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 165)(58 166)(59 167)(60 168)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 161)(68 162)(69 163)(70 164)(85 129)(86 130)(87 131)(88 132)(89 133)(90 134)(91 135)(92 136)(93 137)(94 138)(95 139)(96 140)(97 127)(98 128)(99 169)(100 170)(101 171)(102 172)(103 173)(104 174)(105 175)(106 176)(107 177)(108 178)(109 179)(110 180)(111 181)(112 182)(197 221)(198 222)(199 223)(200 224)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)
(1 192)(2 193)(3 194)(4 195)(5 196)(6 183)(7 184)(8 185)(9 186)(10 187)(11 188)(12 189)(13 190)(14 191)(15 201)(16 202)(17 203)(18 204)(19 205)(20 206)(21 207)(22 208)(23 209)(24 210)(25 197)(26 198)(27 199)(28 200)(29 81)(30 82)(31 83)(32 84)(33 71)(34 72)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 180)(44 181)(45 182)(46 169)(47 170)(48 171)(49 172)(50 173)(51 174)(52 175)(53 176)(54 177)(55 178)(56 179)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(99 124)(100 125)(101 126)(102 113)(103 114)(104 115)(105 116)(106 117)(107 118)(108 119)(109 120)(110 121)(111 122)(112 123)(127 156)(128 157)(129 158)(130 159)(131 160)(132 161)(133 162)(134 163)(135 164)(136 165)(137 166)(138 167)(139 168)(140 155)(141 223)(142 224)(143 211)(144 212)(145 213)(146 214)(147 215)(148 216)(149 217)(150 218)(151 219)(152 220)(153 221)(154 222)
(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 47 39 100)(2 124 40 169)(3 45 41 112)(4 122 42 181)(5 43 29 110)(6 120 30 179)(7 55 31 108)(8 118 32 177)(9 53 33 106)(10 116 34 175)(11 51 35 104)(12 114 36 173)(13 49 37 102)(14 126 38 171)(15 130 211 65)(16 85 212 158)(17 128 213 63)(18 97 214 156)(19 140 215 61)(20 95 216 168)(21 138 217 59)(22 93 218 166)(23 136 219 57)(24 91 220 164)(25 134 221 69)(26 89 222 162)(27 132 223 67)(28 87 224 160)(44 195 111 80)(46 193 99 78)(48 191 101 76)(50 189 103 74)(52 187 105 72)(54 185 107 84)(56 183 109 82)(58 150 137 208)(60 148 139 206)(62 146 127 204)(64 144 129 202)(66 142 131 200)(68 154 133 198)(70 152 135 210)(71 117 186 176)(73 115 188 174)(75 113 190 172)(77 125 192 170)(79 123 194 182)(81 121 196 180)(83 119 184 178)(86 201 159 143)(88 199 161 141)(90 197 163 153)(92 209 165 151)(94 207 167 149)(96 205 155 147)(98 203 157 145)
G:=sub<Sym(224)| (1,153)(2,154)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,81)(16,82)(17,83)(18,84)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,201)(30,202)(31,203)(32,204)(33,205)(34,206)(35,207)(36,208)(37,209)(38,210)(39,197)(40,198)(41,199)(42,200)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,169)(69,170)(70,171)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,136)(114,137)(115,138)(116,139)(117,140)(118,127)(119,128)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,221)(193,222)(194,223)(195,224)(196,211), (1,25)(2,198)(3,27)(4,200)(5,15)(6,202)(7,17)(8,204)(9,19)(10,206)(11,21)(12,208)(13,23)(14,210)(16,183)(18,185)(20,187)(22,189)(24,191)(26,193)(28,195)(29,211)(30,144)(31,213)(32,146)(33,215)(34,148)(35,217)(36,150)(37,219)(38,152)(39,221)(40,154)(41,223)(42,142)(43,159)(44,131)(45,161)(46,133)(47,163)(48,135)(49,165)(50,137)(51,167)(52,139)(53,155)(54,127)(55,157)(56,129)(57,113)(58,103)(59,115)(60,105)(61,117)(62,107)(63,119)(64,109)(65,121)(66,111)(67,123)(68,99)(69,125)(70,101)(71,147)(72,216)(73,149)(74,218)(75,151)(76,220)(77,153)(78,222)(79,141)(80,224)(81,143)(82,212)(83,145)(84,214)(85,120)(86,110)(87,122)(88,112)(89,124)(90,100)(91,126)(92,102)(93,114)(94,104)(95,116)(96,106)(97,118)(98,108)(128,178)(130,180)(132,182)(134,170)(136,172)(138,174)(140,176)(156,177)(158,179)(160,181)(162,169)(164,171)(166,173)(168,175)(184,203)(186,205)(188,207)(190,209)(192,197)(194,199)(196,201), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,141)(28,142)(29,196)(30,183)(31,184)(32,185)(33,186)(34,187)(35,188)(36,189)(37,190)(38,191)(39,192)(40,193)(41,194)(42,195)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,165)(58,166)(59,167)(60,168)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,127)(98,128)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,181)(112,182)(197,221)(198,222)(199,223)(200,224)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220), (1,192)(2,193)(3,194)(4,195)(5,196)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,201)(16,202)(17,203)(18,204)(19,205)(20,206)(21,207)(22,208)(23,209)(24,210)(25,197)(26,198)(27,199)(28,200)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,180)(44,181)(45,182)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(99,124)(100,125)(101,126)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)(111,122)(112,123)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,155)(141,223)(142,224)(143,211)(144,212)(145,213)(146,214)(147,215)(148,216)(149,217)(150,218)(151,219)(152,220)(153,221)(154,222), (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), (1,47,39,100)(2,124,40,169)(3,45,41,112)(4,122,42,181)(5,43,29,110)(6,120,30,179)(7,55,31,108)(8,118,32,177)(9,53,33,106)(10,116,34,175)(11,51,35,104)(12,114,36,173)(13,49,37,102)(14,126,38,171)(15,130,211,65)(16,85,212,158)(17,128,213,63)(18,97,214,156)(19,140,215,61)(20,95,216,168)(21,138,217,59)(22,93,218,166)(23,136,219,57)(24,91,220,164)(25,134,221,69)(26,89,222,162)(27,132,223,67)(28,87,224,160)(44,195,111,80)(46,193,99,78)(48,191,101,76)(50,189,103,74)(52,187,105,72)(54,185,107,84)(56,183,109,82)(58,150,137,208)(60,148,139,206)(62,146,127,204)(64,144,129,202)(66,142,131,200)(68,154,133,198)(70,152,135,210)(71,117,186,176)(73,115,188,174)(75,113,190,172)(77,125,192,170)(79,123,194,182)(81,121,196,180)(83,119,184,178)(86,201,159,143)(88,199,161,141)(90,197,163,153)(92,209,165,151)(94,207,167,149)(96,205,155,147)(98,203,157,145)>;
G:=Group( (1,153)(2,154)(3,141)(4,142)(5,143)(6,144)(7,145)(8,146)(9,147)(10,148)(11,149)(12,150)(13,151)(14,152)(15,81)(16,82)(17,83)(18,84)(19,71)(20,72)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,201)(30,202)(31,203)(32,204)(33,205)(34,206)(35,207)(36,208)(37,209)(38,210)(39,197)(40,198)(41,199)(42,200)(43,86)(44,87)(45,88)(46,89)(47,90)(48,91)(49,92)(50,93)(51,94)(52,95)(53,96)(54,97)(55,98)(56,85)(57,172)(58,173)(59,174)(60,175)(61,176)(62,177)(63,178)(64,179)(65,180)(66,181)(67,182)(68,169)(69,170)(70,171)(99,162)(100,163)(101,164)(102,165)(103,166)(104,167)(105,168)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,161)(113,136)(114,137)(115,138)(116,139)(117,140)(118,127)(119,128)(120,129)(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(183,212)(184,213)(185,214)(186,215)(187,216)(188,217)(189,218)(190,219)(191,220)(192,221)(193,222)(194,223)(195,224)(196,211), (1,25)(2,198)(3,27)(4,200)(5,15)(6,202)(7,17)(8,204)(9,19)(10,206)(11,21)(12,208)(13,23)(14,210)(16,183)(18,185)(20,187)(22,189)(24,191)(26,193)(28,195)(29,211)(30,144)(31,213)(32,146)(33,215)(34,148)(35,217)(36,150)(37,219)(38,152)(39,221)(40,154)(41,223)(42,142)(43,159)(44,131)(45,161)(46,133)(47,163)(48,135)(49,165)(50,137)(51,167)(52,139)(53,155)(54,127)(55,157)(56,129)(57,113)(58,103)(59,115)(60,105)(61,117)(62,107)(63,119)(64,109)(65,121)(66,111)(67,123)(68,99)(69,125)(70,101)(71,147)(72,216)(73,149)(74,218)(75,151)(76,220)(77,153)(78,222)(79,141)(80,224)(81,143)(82,212)(83,145)(84,214)(85,120)(86,110)(87,122)(88,112)(89,124)(90,100)(91,126)(92,102)(93,114)(94,104)(95,116)(96,106)(97,118)(98,108)(128,178)(130,180)(132,182)(134,170)(136,172)(138,174)(140,176)(156,177)(158,179)(160,181)(162,169)(164,171)(166,173)(168,175)(184,203)(186,205)(188,207)(190,209)(192,197)(194,199)(196,201), (1,77)(2,78)(3,79)(4,80)(5,81)(6,82)(7,83)(8,84)(9,71)(10,72)(11,73)(12,74)(13,75)(14,76)(15,143)(16,144)(17,145)(18,146)(19,147)(20,148)(21,149)(22,150)(23,151)(24,152)(25,153)(26,154)(27,141)(28,142)(29,196)(30,183)(31,184)(32,185)(33,186)(34,187)(35,188)(36,189)(37,190)(38,191)(39,192)(40,193)(41,194)(42,195)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,165)(58,166)(59,167)(60,168)(61,155)(62,156)(63,157)(64,158)(65,159)(66,160)(67,161)(68,162)(69,163)(70,164)(85,129)(86,130)(87,131)(88,132)(89,133)(90,134)(91,135)(92,136)(93,137)(94,138)(95,139)(96,140)(97,127)(98,128)(99,169)(100,170)(101,171)(102,172)(103,173)(104,174)(105,175)(106,176)(107,177)(108,178)(109,179)(110,180)(111,181)(112,182)(197,221)(198,222)(199,223)(200,224)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220), (1,192)(2,193)(3,194)(4,195)(5,196)(6,183)(7,184)(8,185)(9,186)(10,187)(11,188)(12,189)(13,190)(14,191)(15,201)(16,202)(17,203)(18,204)(19,205)(20,206)(21,207)(22,208)(23,209)(24,210)(25,197)(26,198)(27,199)(28,200)(29,81)(30,82)(31,83)(32,84)(33,71)(34,72)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78)(41,79)(42,80)(43,180)(44,181)(45,182)(46,169)(47,170)(48,171)(49,172)(50,173)(51,174)(52,175)(53,176)(54,177)(55,178)(56,179)(57,92)(58,93)(59,94)(60,95)(61,96)(62,97)(63,98)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(99,124)(100,125)(101,126)(102,113)(103,114)(104,115)(105,116)(106,117)(107,118)(108,119)(109,120)(110,121)(111,122)(112,123)(127,156)(128,157)(129,158)(130,159)(131,160)(132,161)(133,162)(134,163)(135,164)(136,165)(137,166)(138,167)(139,168)(140,155)(141,223)(142,224)(143,211)(144,212)(145,213)(146,214)(147,215)(148,216)(149,217)(150,218)(151,219)(152,220)(153,221)(154,222), (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), (1,47,39,100)(2,124,40,169)(3,45,41,112)(4,122,42,181)(5,43,29,110)(6,120,30,179)(7,55,31,108)(8,118,32,177)(9,53,33,106)(10,116,34,175)(11,51,35,104)(12,114,36,173)(13,49,37,102)(14,126,38,171)(15,130,211,65)(16,85,212,158)(17,128,213,63)(18,97,214,156)(19,140,215,61)(20,95,216,168)(21,138,217,59)(22,93,218,166)(23,136,219,57)(24,91,220,164)(25,134,221,69)(26,89,222,162)(27,132,223,67)(28,87,224,160)(44,195,111,80)(46,193,99,78)(48,191,101,76)(50,189,103,74)(52,187,105,72)(54,185,107,84)(56,183,109,82)(58,150,137,208)(60,148,139,206)(62,146,127,204)(64,144,129,202)(66,142,131,200)(68,154,133,198)(70,152,135,210)(71,117,186,176)(73,115,188,174)(75,113,190,172)(77,125,192,170)(79,123,194,182)(81,121,196,180)(83,119,184,178)(86,201,159,143)(88,199,161,141)(90,197,163,153)(92,209,165,151)(94,207,167,149)(96,205,155,147)(98,203,157,145) );
G=PermutationGroup([[(1,153),(2,154),(3,141),(4,142),(5,143),(6,144),(7,145),(8,146),(9,147),(10,148),(11,149),(12,150),(13,151),(14,152),(15,81),(16,82),(17,83),(18,84),(19,71),(20,72),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,201),(30,202),(31,203),(32,204),(33,205),(34,206),(35,207),(36,208),(37,209),(38,210),(39,197),(40,198),(41,199),(42,200),(43,86),(44,87),(45,88),(46,89),(47,90),(48,91),(49,92),(50,93),(51,94),(52,95),(53,96),(54,97),(55,98),(56,85),(57,172),(58,173),(59,174),(60,175),(61,176),(62,177),(63,178),(64,179),(65,180),(66,181),(67,182),(68,169),(69,170),(70,171),(99,162),(100,163),(101,164),(102,165),(103,166),(104,167),(105,168),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,161),(113,136),(114,137),(115,138),(116,139),(117,140),(118,127),(119,128),(120,129),(121,130),(122,131),(123,132),(124,133),(125,134),(126,135),(183,212),(184,213),(185,214),(186,215),(187,216),(188,217),(189,218),(190,219),(191,220),(192,221),(193,222),(194,223),(195,224),(196,211)], [(1,25),(2,198),(3,27),(4,200),(5,15),(6,202),(7,17),(8,204),(9,19),(10,206),(11,21),(12,208),(13,23),(14,210),(16,183),(18,185),(20,187),(22,189),(24,191),(26,193),(28,195),(29,211),(30,144),(31,213),(32,146),(33,215),(34,148),(35,217),(36,150),(37,219),(38,152),(39,221),(40,154),(41,223),(42,142),(43,159),(44,131),(45,161),(46,133),(47,163),(48,135),(49,165),(50,137),(51,167),(52,139),(53,155),(54,127),(55,157),(56,129),(57,113),(58,103),(59,115),(60,105),(61,117),(62,107),(63,119),(64,109),(65,121),(66,111),(67,123),(68,99),(69,125),(70,101),(71,147),(72,216),(73,149),(74,218),(75,151),(76,220),(77,153),(78,222),(79,141),(80,224),(81,143),(82,212),(83,145),(84,214),(85,120),(86,110),(87,122),(88,112),(89,124),(90,100),(91,126),(92,102),(93,114),(94,104),(95,116),(96,106),(97,118),(98,108),(128,178),(130,180),(132,182),(134,170),(136,172),(138,174),(140,176),(156,177),(158,179),(160,181),(162,169),(164,171),(166,173),(168,175),(184,203),(186,205),(188,207),(190,209),(192,197),(194,199),(196,201)], [(1,77),(2,78),(3,79),(4,80),(5,81),(6,82),(7,83),(8,84),(9,71),(10,72),(11,73),(12,74),(13,75),(14,76),(15,143),(16,144),(17,145),(18,146),(19,147),(20,148),(21,149),(22,150),(23,151),(24,152),(25,153),(26,154),(27,141),(28,142),(29,196),(30,183),(31,184),(32,185),(33,186),(34,187),(35,188),(36,189),(37,190),(38,191),(39,192),(40,193),(41,194),(42,195),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,165),(58,166),(59,167),(60,168),(61,155),(62,156),(63,157),(64,158),(65,159),(66,160),(67,161),(68,162),(69,163),(70,164),(85,129),(86,130),(87,131),(88,132),(89,133),(90,134),(91,135),(92,136),(93,137),(94,138),(95,139),(96,140),(97,127),(98,128),(99,169),(100,170),(101,171),(102,172),(103,173),(104,174),(105,175),(106,176),(107,177),(108,178),(109,179),(110,180),(111,181),(112,182),(197,221),(198,222),(199,223),(200,224),(201,211),(202,212),(203,213),(204,214),(205,215),(206,216),(207,217),(208,218),(209,219),(210,220)], [(1,192),(2,193),(3,194),(4,195),(5,196),(6,183),(7,184),(8,185),(9,186),(10,187),(11,188),(12,189),(13,190),(14,191),(15,201),(16,202),(17,203),(18,204),(19,205),(20,206),(21,207),(22,208),(23,209),(24,210),(25,197),(26,198),(27,199),(28,200),(29,81),(30,82),(31,83),(32,84),(33,71),(34,72),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78),(41,79),(42,80),(43,180),(44,181),(45,182),(46,169),(47,170),(48,171),(49,172),(50,173),(51,174),(52,175),(53,176),(54,177),(55,178),(56,179),(57,92),(58,93),(59,94),(60,95),(61,96),(62,97),(63,98),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(99,124),(100,125),(101,126),(102,113),(103,114),(104,115),(105,116),(106,117),(107,118),(108,119),(109,120),(110,121),(111,122),(112,123),(127,156),(128,157),(129,158),(130,159),(131,160),(132,161),(133,162),(134,163),(135,164),(136,165),(137,166),(138,167),(139,168),(140,155),(141,223),(142,224),(143,211),(144,212),(145,213),(146,214),(147,215),(148,216),(149,217),(150,218),(151,219),(152,220),(153,221),(154,222)], [(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)], [(1,47,39,100),(2,124,40,169),(3,45,41,112),(4,122,42,181),(5,43,29,110),(6,120,30,179),(7,55,31,108),(8,118,32,177),(9,53,33,106),(10,116,34,175),(11,51,35,104),(12,114,36,173),(13,49,37,102),(14,126,38,171),(15,130,211,65),(16,85,212,158),(17,128,213,63),(18,97,214,156),(19,140,215,61),(20,95,216,168),(21,138,217,59),(22,93,218,166),(23,136,219,57),(24,91,220,164),(25,134,221,69),(26,89,222,162),(27,132,223,67),(28,87,224,160),(44,195,111,80),(46,193,99,78),(48,191,101,76),(50,189,103,74),(52,187,105,72),(54,185,107,84),(56,183,109,82),(58,150,137,208),(60,148,139,206),(62,146,127,204),(64,144,129,202),(66,142,131,200),(68,154,133,198),(70,152,135,210),(71,117,186,176),(73,115,188,174),(75,113,190,172),(77,125,192,170),(79,123,194,182),(81,121,196,180),(83,119,184,178),(86,201,159,143),(88,199,161,141),(90,197,163,153),(92,209,165,151),(94,207,167,149),(96,205,155,147),(98,203,157,145)]])
88 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | 4B | 4C | ··· | 4J | 4K | 4L | 4M | 4N | 7A | 7B | 7C | 14A | ··· | 14U | 14V | ··· | 14AS | 28A | ··· | 28L |
| 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 | 4 | 4 | 4 | 4 | 14 | ··· | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
88 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C7⋊D4 | D4⋊2D7 |
| kernel | C2×C23.18D14 | C2×Dic7⋊C4 | C23.18D14 | C2×C23.D7 | C23×Dic7 | D4×C2×C14 | C22×C14 | C22×D4 | C2×C14 | C22×C4 | C2×D4 | C24 | C23 | C22 |
| # reps | 1 | 2 | 8 | 3 | 1 | 1 | 4 | 3 | 8 | 3 | 12 | 6 | 24 | 12 |
Matrix representation of C2×C23.18D14 ►in GL6(𝔽29)
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 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 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 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 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 25 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 15 | 9 | 0 | 0 | 0 | 0 |
| 20 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 17 |
| 0 | 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[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],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,22,22,0,0,0,0,25,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[15,20,0,0,0,0,9,14,0,0,0,0,0,0,28,0,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,17,0] >;
C2×C23.18D14 in GAP, Magma, Sage, TeX
C_2\times C_2^3._{18}D_{14} % in TeX
G:=Group("C2xC2^3.18D14"); // GroupNames label
G:=SmallGroup(448,1249);
// by ID
G=gap.SmallGroup(448,1249);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,675,297,18822]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,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,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations