direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8⋊3D7, D8⋊12D14, C28.3C24, C56.32C23, Dic28⋊13C22, Dic14.1C23, (C14×D8)⋊8C2, (C2×D8)⋊13D7, C4.41(D4×D7), C14⋊2(C4○D8), D14.9(C2×D4), (C4×D7).27D4, C28.78(C2×D4), C7⋊C8.20C23, C4.3(C23×D7), (C2×C8).245D14, (C7×D8)⋊10C22, (C8×D7)⋊14C22, D4.D7⋊8C22, (C7×D4).1C23, D4.1(C22×D7), C8.38(C22×D7), (C2×Dic28)⋊19C2, (C2×D4).180D14, D4⋊2D7⋊6C22, (C2×C56).97C22, Dic7.68(C2×D4), (C22×D7).61D4, (C4×D7).24C23, C22.137(D4×D7), (C2×C28).520C23, (C2×Dic7).215D4, C14.104(C22×D4), (D4×C14).162C22, (C2×Dic14).195C22, (D7×C2×C8)⋊5C2, C7⋊2(C2×C4○D8), C2.77(C2×D4×D7), (C2×D4.D7)⋊26C2, (C2×D4⋊2D7)⋊24C2, (C2×C14).393(C2×D4), (C2×C7⋊C8).283C22, (C2×C4×D7).256C22, (C2×C4).610(C22×D7), SmallGroup(448,1209)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1188 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×12], C7, C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×4], D4 [×10], Q8 [×6], C23 [×3], D7 [×2], C14, C14 [×2], C14 [×4], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×12], Dic7 [×2], Dic7 [×4], C28 [×2], D14 [×2], D14 [×2], C2×C14, C2×C14 [×8], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C7⋊C8 [×2], C56 [×2], Dic14 [×4], Dic14 [×2], C4×D7 [×4], C2×Dic7, C2×Dic7 [×10], C7⋊D4 [×8], C2×C28, C7×D4 [×4], C7×D4 [×2], C22×D7, C22×C14 [×2], C2×C4○D8, C8×D7 [×4], Dic28 [×4], C2×C7⋊C8, D4.D7 [×8], C2×C56, C7×D8 [×4], C2×Dic14 [×2], C2×C4×D7, D4⋊2D7 [×8], D4⋊2D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×2], D4×C14 [×2], D7×C2×C8, C2×Dic28, D8⋊3D7 [×8], C2×D4.D7 [×2], C14×D8, C2×D4⋊2D7 [×2], C2×D8⋊3D7
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D7, C2×D4 [×6], C24, D14 [×7], C4○D8 [×2], C22×D4, C22×D7 [×7], C2×C4○D8, D4×D7 [×2], C23×D7, D8⋊3D7 [×2], C2×D4×D7, C2×D8⋊3D7
Generators and relations
G = < a,b,c,d,e | a2=b8=c2=d7=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >
►On 224 points
Generators in S224
(1 196)(2 197)(3 198)(4 199)(5 200)(6 193)(7 194)(8 195)(9 166)(10 167)(11 168)(12 161)(13 162)(14 163)(15 164)(16 165)(17 160)(18 153)(19 154)(20 155)(21 156)(22 157)(23 158)(24 159)(25 221)(26 222)(27 223)(28 224)(29 217)(30 218)(31 219)(32 220)(33 136)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 214)(42 215)(43 216)(44 209)(45 210)(46 211)(47 212)(48 213)(49 208)(50 201)(51 202)(52 203)(53 204)(54 205)(55 206)(56 207)(57 124)(58 125)(59 126)(60 127)(61 128)(62 121)(63 122)(64 123)(65 118)(66 119)(67 120)(68 113)(69 114)(70 115)(71 116)(72 117)(73 141)(74 142)(75 143)(76 144)(77 137)(78 138)(79 139)(80 140)(81 175)(82 176)(83 169)(84 170)(85 171)(86 172)(87 173)(88 174)(89 183)(90 184)(91 177)(92 178)(93 179)(94 180)(95 181)(96 182)(97 190)(98 191)(99 192)(100 185)(101 186)(102 187)(103 188)(104 189)(105 149)(106 150)(107 151)(108 152)(109 145)(110 146)(111 147)(112 148)
(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)
(1 221)(2 220)(3 219)(4 218)(5 217)(6 224)(7 223)(8 222)(9 51)(10 50)(11 49)(12 56)(13 55)(14 54)(15 53)(16 52)(17 43)(18 42)(19 41)(20 48)(21 47)(22 46)(23 45)(24 44)(25 196)(26 195)(27 194)(28 193)(29 200)(30 199)(31 198)(32 197)(33 112)(34 111)(35 110)(36 109)(37 108)(38 107)(39 106)(40 105)(57 88)(58 87)(59 86)(60 85)(61 84)(62 83)(63 82)(64 81)(65 94)(66 93)(67 92)(68 91)(69 90)(70 89)(71 96)(72 95)(73 99)(74 98)(75 97)(76 104)(77 103)(78 102)(79 101)(80 100)(113 177)(114 184)(115 183)(116 182)(117 181)(118 180)(119 179)(120 178)(121 169)(122 176)(123 175)(124 174)(125 173)(126 172)(127 171)(128 170)(129 147)(130 146)(131 145)(132 152)(133 151)(134 150)(135 149)(136 148)(137 188)(138 187)(139 186)(140 185)(141 192)(142 191)(143 190)(144 189)(153 215)(154 214)(155 213)(156 212)(157 211)(158 210)(159 209)(160 216)(161 207)(162 206)(163 205)(164 204)(165 203)(166 202)(167 201)(168 208)
(1 57 47 151 183 163 187)(2 58 48 152 184 164 188)(3 59 41 145 177 165 189)(4 60 42 146 178 166 190)(5 61 43 147 179 167 191)(6 62 44 148 180 168 192)(7 63 45 149 181 161 185)(8 64 46 150 182 162 186)(9 97 199 127 215 110 92)(10 98 200 128 216 111 93)(11 99 193 121 209 112 94)(12 100 194 122 210 105 95)(13 101 195 123 211 106 96)(14 102 196 124 212 107 89)(15 103 197 125 213 108 90)(16 104 198 126 214 109 91)(17 129 119 201 142 217 84)(18 130 120 202 143 218 85)(19 131 113 203 144 219 86)(20 132 114 204 137 220 87)(21 133 115 205 138 221 88)(22 134 116 206 139 222 81)(23 135 117 207 140 223 82)(24 136 118 208 141 224 83)(25 174 156 38 70 54 78)(26 175 157 39 71 55 79)(27 176 158 40 72 56 80)(28 169 159 33 65 49 73)(29 170 160 34 66 50 74)(30 171 153 35 67 51 75)(31 172 154 36 68 52 76)(32 173 155 37 69 53 77)
(1 102)(2 103)(3 104)(4 97)(5 98)(6 99)(7 100)(8 101)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 70)(18 71)(19 72)(20 65)(21 66)(22 67)(23 68)(24 69)(25 142)(26 143)(27 144)(28 137)(29 138)(30 139)(31 140)(32 141)(33 132)(34 133)(35 134)(36 135)(37 136)(38 129)(39 130)(40 131)(41 91)(42 92)(43 93)(44 94)(45 95)(46 96)(47 89)(48 90)(49 87)(50 88)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(73 220)(74 221)(75 222)(76 223)(77 224)(78 217)(79 218)(80 219)(105 149)(106 150)(107 151)(108 152)(109 145)(110 146)(111 147)(112 148)(113 158)(114 159)(115 160)(116 153)(117 154)(118 155)(119 156)(120 157)(121 168)(122 161)(123 162)(124 163)(125 164)(126 165)(127 166)(128 167)(169 204)(170 205)(171 206)(172 207)(173 208)(174 201)(175 202)(176 203)(177 214)(178 215)(179 216)(180 209)(181 210)(182 211)(183 212)(184 213)(185 194)(186 195)(187 196)(188 197)(189 198)(190 199)(191 200)(192 193)
G:=sub<Sym(224)| (1,196)(2,197)(3,198)(4,199)(5,200)(6,193)(7,194)(8,195)(9,166)(10,167)(11,168)(12,161)(13,162)(14,163)(15,164)(16,165)(17,160)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,136)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,214)(42,215)(43,216)(44,209)(45,210)(46,211)(47,212)(48,213)(49,208)(50,201)(51,202)(52,203)(53,204)(54,205)(55,206)(56,207)(57,124)(58,125)(59,126)(60,127)(61,128)(62,121)(63,122)(64,123)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,141)(74,142)(75,143)(76,144)(77,137)(78,138)(79,139)(80,140)(81,175)(82,176)(83,169)(84,170)(85,171)(86,172)(87,173)(88,174)(89,183)(90,184)(91,177)(92,178)(93,179)(94,180)(95,181)(96,182)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148), (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), (1,221)(2,220)(3,219)(4,218)(5,217)(6,224)(7,223)(8,222)(9,51)(10,50)(11,49)(12,56)(13,55)(14,54)(15,53)(16,52)(17,43)(18,42)(19,41)(20,48)(21,47)(22,46)(23,45)(24,44)(25,196)(26,195)(27,194)(28,193)(29,200)(30,199)(31,198)(32,197)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,96)(72,95)(73,99)(74,98)(75,97)(76,104)(77,103)(78,102)(79,101)(80,100)(113,177)(114,184)(115,183)(116,182)(117,181)(118,180)(119,179)(120,178)(121,169)(122,176)(123,175)(124,174)(125,173)(126,172)(127,171)(128,170)(129,147)(130,146)(131,145)(132,152)(133,151)(134,150)(135,149)(136,148)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,209)(160,216)(161,207)(162,206)(163,205)(164,204)(165,203)(166,202)(167,201)(168,208), (1,57,47,151,183,163,187)(2,58,48,152,184,164,188)(3,59,41,145,177,165,189)(4,60,42,146,178,166,190)(5,61,43,147,179,167,191)(6,62,44,148,180,168,192)(7,63,45,149,181,161,185)(8,64,46,150,182,162,186)(9,97,199,127,215,110,92)(10,98,200,128,216,111,93)(11,99,193,121,209,112,94)(12,100,194,122,210,105,95)(13,101,195,123,211,106,96)(14,102,196,124,212,107,89)(15,103,197,125,213,108,90)(16,104,198,126,214,109,91)(17,129,119,201,142,217,84)(18,130,120,202,143,218,85)(19,131,113,203,144,219,86)(20,132,114,204,137,220,87)(21,133,115,205,138,221,88)(22,134,116,206,139,222,81)(23,135,117,207,140,223,82)(24,136,118,208,141,224,83)(25,174,156,38,70,54,78)(26,175,157,39,71,55,79)(27,176,158,40,72,56,80)(28,169,159,33,65,49,73)(29,170,160,34,66,50,74)(30,171,153,35,67,51,75)(31,172,154,36,68,52,76)(32,173,155,37,69,53,77), (1,102)(2,103)(3,104)(4,97)(5,98)(6,99)(7,100)(8,101)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,142)(26,143)(27,144)(28,137)(29,138)(30,139)(31,140)(32,141)(33,132)(34,133)(35,134)(36,135)(37,136)(38,129)(39,130)(40,131)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,89)(48,90)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(73,220)(74,221)(75,222)(76,223)(77,224)(78,217)(79,218)(80,219)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(113,158)(114,159)(115,160)(116,153)(117,154)(118,155)(119,156)(120,157)(121,168)(122,161)(123,162)(124,163)(125,164)(126,165)(127,166)(128,167)(169,204)(170,205)(171,206)(172,207)(173,208)(174,201)(175,202)(176,203)(177,214)(178,215)(179,216)(180,209)(181,210)(182,211)(183,212)(184,213)(185,194)(186,195)(187,196)(188,197)(189,198)(190,199)(191,200)(192,193)>;
G:=Group( (1,196)(2,197)(3,198)(4,199)(5,200)(6,193)(7,194)(8,195)(9,166)(10,167)(11,168)(12,161)(13,162)(14,163)(15,164)(16,165)(17,160)(18,153)(19,154)(20,155)(21,156)(22,157)(23,158)(24,159)(25,221)(26,222)(27,223)(28,224)(29,217)(30,218)(31,219)(32,220)(33,136)(34,129)(35,130)(36,131)(37,132)(38,133)(39,134)(40,135)(41,214)(42,215)(43,216)(44,209)(45,210)(46,211)(47,212)(48,213)(49,208)(50,201)(51,202)(52,203)(53,204)(54,205)(55,206)(56,207)(57,124)(58,125)(59,126)(60,127)(61,128)(62,121)(63,122)(64,123)(65,118)(66,119)(67,120)(68,113)(69,114)(70,115)(71,116)(72,117)(73,141)(74,142)(75,143)(76,144)(77,137)(78,138)(79,139)(80,140)(81,175)(82,176)(83,169)(84,170)(85,171)(86,172)(87,173)(88,174)(89,183)(90,184)(91,177)(92,178)(93,179)(94,180)(95,181)(96,182)(97,190)(98,191)(99,192)(100,185)(101,186)(102,187)(103,188)(104,189)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148), (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), (1,221)(2,220)(3,219)(4,218)(5,217)(6,224)(7,223)(8,222)(9,51)(10,50)(11,49)(12,56)(13,55)(14,54)(15,53)(16,52)(17,43)(18,42)(19,41)(20,48)(21,47)(22,46)(23,45)(24,44)(25,196)(26,195)(27,194)(28,193)(29,200)(30,199)(31,198)(32,197)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,96)(72,95)(73,99)(74,98)(75,97)(76,104)(77,103)(78,102)(79,101)(80,100)(113,177)(114,184)(115,183)(116,182)(117,181)(118,180)(119,179)(120,178)(121,169)(122,176)(123,175)(124,174)(125,173)(126,172)(127,171)(128,170)(129,147)(130,146)(131,145)(132,152)(133,151)(134,150)(135,149)(136,148)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189)(153,215)(154,214)(155,213)(156,212)(157,211)(158,210)(159,209)(160,216)(161,207)(162,206)(163,205)(164,204)(165,203)(166,202)(167,201)(168,208), (1,57,47,151,183,163,187)(2,58,48,152,184,164,188)(3,59,41,145,177,165,189)(4,60,42,146,178,166,190)(5,61,43,147,179,167,191)(6,62,44,148,180,168,192)(7,63,45,149,181,161,185)(8,64,46,150,182,162,186)(9,97,199,127,215,110,92)(10,98,200,128,216,111,93)(11,99,193,121,209,112,94)(12,100,194,122,210,105,95)(13,101,195,123,211,106,96)(14,102,196,124,212,107,89)(15,103,197,125,213,108,90)(16,104,198,126,214,109,91)(17,129,119,201,142,217,84)(18,130,120,202,143,218,85)(19,131,113,203,144,219,86)(20,132,114,204,137,220,87)(21,133,115,205,138,221,88)(22,134,116,206,139,222,81)(23,135,117,207,140,223,82)(24,136,118,208,141,224,83)(25,174,156,38,70,54,78)(26,175,157,39,71,55,79)(27,176,158,40,72,56,80)(28,169,159,33,65,49,73)(29,170,160,34,66,50,74)(30,171,153,35,67,51,75)(31,172,154,36,68,52,76)(32,173,155,37,69,53,77), (1,102)(2,103)(3,104)(4,97)(5,98)(6,99)(7,100)(8,101)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,70)(18,71)(19,72)(20,65)(21,66)(22,67)(23,68)(24,69)(25,142)(26,143)(27,144)(28,137)(29,138)(30,139)(31,140)(32,141)(33,132)(34,133)(35,134)(36,135)(37,136)(38,129)(39,130)(40,131)(41,91)(42,92)(43,93)(44,94)(45,95)(46,96)(47,89)(48,90)(49,87)(50,88)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(73,220)(74,221)(75,222)(76,223)(77,224)(78,217)(79,218)(80,219)(105,149)(106,150)(107,151)(108,152)(109,145)(110,146)(111,147)(112,148)(113,158)(114,159)(115,160)(116,153)(117,154)(118,155)(119,156)(120,157)(121,168)(122,161)(123,162)(124,163)(125,164)(126,165)(127,166)(128,167)(169,204)(170,205)(171,206)(172,207)(173,208)(174,201)(175,202)(176,203)(177,214)(178,215)(179,216)(180,209)(181,210)(182,211)(183,212)(184,213)(185,194)(186,195)(187,196)(188,197)(189,198)(190,199)(191,200)(192,193) );
G=PermutationGroup([(1,196),(2,197),(3,198),(4,199),(5,200),(6,193),(7,194),(8,195),(9,166),(10,167),(11,168),(12,161),(13,162),(14,163),(15,164),(16,165),(17,160),(18,153),(19,154),(20,155),(21,156),(22,157),(23,158),(24,159),(25,221),(26,222),(27,223),(28,224),(29,217),(30,218),(31,219),(32,220),(33,136),(34,129),(35,130),(36,131),(37,132),(38,133),(39,134),(40,135),(41,214),(42,215),(43,216),(44,209),(45,210),(46,211),(47,212),(48,213),(49,208),(50,201),(51,202),(52,203),(53,204),(54,205),(55,206),(56,207),(57,124),(58,125),(59,126),(60,127),(61,128),(62,121),(63,122),(64,123),(65,118),(66,119),(67,120),(68,113),(69,114),(70,115),(71,116),(72,117),(73,141),(74,142),(75,143),(76,144),(77,137),(78,138),(79,139),(80,140),(81,175),(82,176),(83,169),(84,170),(85,171),(86,172),(87,173),(88,174),(89,183),(90,184),(91,177),(92,178),(93,179),(94,180),(95,181),(96,182),(97,190),(98,191),(99,192),(100,185),(101,186),(102,187),(103,188),(104,189),(105,149),(106,150),(107,151),(108,152),(109,145),(110,146),(111,147),(112,148)], [(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)], [(1,221),(2,220),(3,219),(4,218),(5,217),(6,224),(7,223),(8,222),(9,51),(10,50),(11,49),(12,56),(13,55),(14,54),(15,53),(16,52),(17,43),(18,42),(19,41),(20,48),(21,47),(22,46),(23,45),(24,44),(25,196),(26,195),(27,194),(28,193),(29,200),(30,199),(31,198),(32,197),(33,112),(34,111),(35,110),(36,109),(37,108),(38,107),(39,106),(40,105),(57,88),(58,87),(59,86),(60,85),(61,84),(62,83),(63,82),(64,81),(65,94),(66,93),(67,92),(68,91),(69,90),(70,89),(71,96),(72,95),(73,99),(74,98),(75,97),(76,104),(77,103),(78,102),(79,101),(80,100),(113,177),(114,184),(115,183),(116,182),(117,181),(118,180),(119,179),(120,178),(121,169),(122,176),(123,175),(124,174),(125,173),(126,172),(127,171),(128,170),(129,147),(130,146),(131,145),(132,152),(133,151),(134,150),(135,149),(136,148),(137,188),(138,187),(139,186),(140,185),(141,192),(142,191),(143,190),(144,189),(153,215),(154,214),(155,213),(156,212),(157,211),(158,210),(159,209),(160,216),(161,207),(162,206),(163,205),(164,204),(165,203),(166,202),(167,201),(168,208)], [(1,57,47,151,183,163,187),(2,58,48,152,184,164,188),(3,59,41,145,177,165,189),(4,60,42,146,178,166,190),(5,61,43,147,179,167,191),(6,62,44,148,180,168,192),(7,63,45,149,181,161,185),(8,64,46,150,182,162,186),(9,97,199,127,215,110,92),(10,98,200,128,216,111,93),(11,99,193,121,209,112,94),(12,100,194,122,210,105,95),(13,101,195,123,211,106,96),(14,102,196,124,212,107,89),(15,103,197,125,213,108,90),(16,104,198,126,214,109,91),(17,129,119,201,142,217,84),(18,130,120,202,143,218,85),(19,131,113,203,144,219,86),(20,132,114,204,137,220,87),(21,133,115,205,138,221,88),(22,134,116,206,139,222,81),(23,135,117,207,140,223,82),(24,136,118,208,141,224,83),(25,174,156,38,70,54,78),(26,175,157,39,71,55,79),(27,176,158,40,72,56,80),(28,169,159,33,65,49,73),(29,170,160,34,66,50,74),(30,171,153,35,67,51,75),(31,172,154,36,68,52,76),(32,173,155,37,69,53,77)], [(1,102),(2,103),(3,104),(4,97),(5,98),(6,99),(7,100),(8,101),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,70),(18,71),(19,72),(20,65),(21,66),(22,67),(23,68),(24,69),(25,142),(26,143),(27,144),(28,137),(29,138),(30,139),(31,140),(32,141),(33,132),(34,133),(35,134),(36,135),(37,136),(38,129),(39,130),(40,131),(41,91),(42,92),(43,93),(44,94),(45,95),(46,96),(47,89),(48,90),(49,87),(50,88),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(73,220),(74,221),(75,222),(76,223),(77,224),(78,217),(79,218),(80,219),(105,149),(106,150),(107,151),(108,152),(109,145),(110,146),(111,147),(112,148),(113,158),(114,159),(115,160),(116,153),(117,154),(118,155),(119,156),(120,157),(121,168),(122,161),(123,162),(124,163),(125,164),(126,165),(127,166),(128,167),(169,204),(170,205),(171,206),(172,207),(173,208),(174,201),(175,202),(176,203),(177,214),(178,215),(179,216),(180,209),(181,210),(182,211),(183,212),(184,213),(185,194),(186,195),(187,196),(188,197),(189,198),(190,199),(191,200),(192,193)])
Matrix representation ►G ⊆ GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 32 | 44 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 111 | 69 |
| 0 | 0 | 54 | 2 |
| 9 | 112 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 104 | 1 | 0 | 0 |
| 33 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 41 | 112 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,18,32,0,0,0,44],[1,0,0,0,0,1,0,0,0,0,111,54,0,0,69,2],[9,1,0,0,112,0,0,0,0,0,1,0,0,0,0,1],[104,33,0,0,1,9,0,0,0,0,1,41,0,0,0,112] >;
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 14J | ··· | 14U | 28A | ··· | 28F | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 14 | 14 | 2 | 2 | 7 | 7 | 7 | 7 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 4 | ··· | 4 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D7 | D14 | D14 | D14 | C4○D8 | D4×D7 | D4×D7 | D8⋊3D7 |
| kernel | C2×D8⋊3D7 | D7×C2×C8 | C2×Dic28 | D8⋊3D7 | C2×D4.D7 | C14×D8 | C2×D4⋊2D7 | C4×D7 | C2×Dic7 | C22×D7 | C2×D8 | C2×C8 | D8 | C2×D4 | C14 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 1 | 1 | 3 | 3 | 12 | 6 | 8 | 3 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_2\times D_8\rtimes_3D_7
% in TeX
G:=Group("C2xD8:3D7"); // GroupNames label
G:=SmallGroup(448,1209);
// by ID
G=gap.SmallGroup(448,1209);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,1123,185,438,235,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀