direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D4.Dic7, C28.75C24, C14⋊3(C8○D4), C7⋊C8.37C23, C4○D4.44D14, (D4×C14).12C4, (Q8×C14).12C4, C4○D4.4Dic7, D4.8(C2×Dic7), Q8.9(C2×Dic7), C4.74(C23×D7), C28.97(C22×C4), C14.48(C23×C4), (C2×D4).12Dic7, (C2×Q8).10Dic7, (C2×C28).553C23, (C22×C4).386D14, C4.Dic7⋊34C22, C2.10(C23×Dic7), C4.40(C22×Dic7), C23.19(C2×Dic7), (C22×C28).288C22, C22.33(C22×Dic7), C7⋊4(C2×C8○D4), (C2×C7⋊C8)⋊41C22, (C22×C7⋊C8)⋊14C2, (C7×C4○D4).3C4, (C14×C4○D4).9C2, (C2×C4○D4).13D7, (C7×D4).25(C2×C4), (C7×Q8).27(C2×C4), (C2×C28).135(C2×C4), (C2×C4.Dic7)⋊28C2, (C2×C4).56(C2×Dic7), (C22×C14).80(C2×C4), (C2×C14).28(C22×C4), (C7×C4○D4).48C22, (C2×C4).831(C22×D7), SmallGroup(448,1271)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D4.Dic7
G = < a,b,c,d,e | a2=b4=c2=1, d14=b2, e2=b2d7, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d13 >
Subgroups: 596 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C7, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C14, C14 [×2], C14 [×6], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C28 [×2], C28 [×6], C2×C14, C2×C14 [×6], C2×C14 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C7⋊C8 [×8], C2×C28, C2×C28 [×15], C7×D4 [×12], C7×Q8 [×4], C22×C14 [×3], C2×C8○D4, C2×C7⋊C8, C2×C7⋊C8 [×15], C4.Dic7 [×12], C22×C28 [×3], D4×C14 [×3], Q8×C14, C7×C4○D4 [×8], C22×C7⋊C8 [×3], C2×C4.Dic7 [×3], D4.Dic7 [×8], C14×C4○D4, C2×D4.Dic7
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D7, C22×C4 [×14], C24, Dic7 [×8], D14 [×7], C8○D4 [×2], C23×C4, C2×Dic7 [×28], C22×D7 [×7], C2×C8○D4, C22×Dic7 [×14], C23×D7, D4.Dic7 [×2], C23×Dic7, C2×D4.Dic7
►On 224 points
Generators in S224
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)(113 146)(114 147)(115 148)(116 149)(117 150)(118 151)(119 152)(120 153)(121 154)(122 155)(123 156)(124 157)(125 158)(126 159)(127 160)(128 161)(129 162)(130 163)(131 164)(132 165)(133 166)(134 167)(135 168)(136 141)(137 142)(138 143)(139 144)(140 145)(169 215)(170 216)(171 217)(172 218)(173 219)(174 220)(175 221)(176 222)(177 223)(178 224)(179 197)(180 198)(181 199)(182 200)(183 201)(184 202)(185 203)(186 204)(187 205)(188 206)(189 207)(190 208)(191 209)(192 210)(193 211)(194 212)(195 213)(196 214)
(1 22 15 8)(2 23 16 9)(3 24 17 10)(4 25 18 11)(5 26 19 12)(6 27 20 13)(7 28 21 14)(29 50 43 36)(30 51 44 37)(31 52 45 38)(32 53 46 39)(33 54 47 40)(34 55 48 41)(35 56 49 42)(57 64 71 78)(58 65 72 79)(59 66 73 80)(60 67 74 81)(61 68 75 82)(62 69 76 83)(63 70 77 84)(85 92 99 106)(86 93 100 107)(87 94 101 108)(88 95 102 109)(89 96 103 110)(90 97 104 111)(91 98 105 112)(113 134 127 120)(114 135 128 121)(115 136 129 122)(116 137 130 123)(117 138 131 124)(118 139 132 125)(119 140 133 126)(141 162 155 148)(142 163 156 149)(143 164 157 150)(144 165 158 151)(145 166 159 152)(146 167 160 153)(147 168 161 154)(169 176 183 190)(170 177 184 191)(171 178 185 192)(172 179 186 193)(173 180 187 194)(174 181 188 195)(175 182 189 196)(197 204 211 218)(198 205 212 219)(199 206 213 220)(200 207 214 221)(201 208 215 222)(202 209 216 223)(203 210 217 224)
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)(113 183)(114 184)(115 185)(116 186)(117 187)(118 188)(119 189)(120 190)(121 191)(122 192)(123 193)(124 194)(125 195)(126 196)(127 169)(128 170)(129 171)(130 172)(131 173)(132 174)(133 175)(134 176)(135 177)(136 178)(137 179)(138 180)(139 181)(140 182)(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 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)
(1 154 22 147 15 168 8 161)(2 167 23 160 16 153 9 146)(3 152 24 145 17 166 10 159)(4 165 25 158 18 151 11 144)(5 150 26 143 19 164 12 157)(6 163 27 156 20 149 13 142)(7 148 28 141 21 162 14 155)(29 121 50 114 43 135 36 128)(30 134 51 127 44 120 37 113)(31 119 52 140 45 133 38 126)(32 132 53 125 46 118 39 139)(33 117 54 138 47 131 40 124)(34 130 55 123 48 116 41 137)(35 115 56 136 49 129 42 122)(57 209 78 202 71 223 64 216)(58 222 79 215 72 208 65 201)(59 207 80 200 73 221 66 214)(60 220 81 213 74 206 67 199)(61 205 82 198 75 219 68 212)(62 218 83 211 76 204 69 197)(63 203 84 224 77 217 70 210)(85 191 106 184 99 177 92 170)(86 176 107 169 100 190 93 183)(87 189 108 182 101 175 94 196)(88 174 109 195 102 188 95 181)(89 187 110 180 103 173 96 194)(90 172 111 193 104 186 97 179)(91 185 112 178 105 171 98 192)
G:=sub<Sym(224)| (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153)(121,154)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,161)(129,162)(130,163)(131,164)(132,165)(133,166)(134,167)(135,168)(136,141)(137,142)(138,143)(139,144)(140,145)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,197)(180,198)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,211)(194,212)(195,213)(196,214), (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)(113,134,127,120)(114,135,128,121)(115,136,129,122)(116,137,130,123)(117,138,131,124)(118,139,132,125)(119,140,133,126)(141,162,155,148)(142,163,156,149)(143,164,157,150)(144,165,158,151)(145,166,159,152)(146,167,160,153)(147,168,161,154)(169,176,183,190)(170,177,184,191)(171,178,185,192)(172,179,186,193)(173,180,187,194)(174,181,188,195)(175,182,189,196)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,154,22,147,15,168,8,161)(2,167,23,160,16,153,9,146)(3,152,24,145,17,166,10,159)(4,165,25,158,18,151,11,144)(5,150,26,143,19,164,12,157)(6,163,27,156,20,149,13,142)(7,148,28,141,21,162,14,155)(29,121,50,114,43,135,36,128)(30,134,51,127,44,120,37,113)(31,119,52,140,45,133,38,126)(32,132,53,125,46,118,39,139)(33,117,54,138,47,131,40,124)(34,130,55,123,48,116,41,137)(35,115,56,136,49,129,42,122)(57,209,78,202,71,223,64,216)(58,222,79,215,72,208,65,201)(59,207,80,200,73,221,66,214)(60,220,81,213,74,206,67,199)(61,205,82,198,75,219,68,212)(62,218,83,211,76,204,69,197)(63,203,84,224,77,217,70,210)(85,191,106,184,99,177,92,170)(86,176,107,169,100,190,93,183)(87,189,108,182,101,175,94,196)(88,174,109,195,102,188,95,181)(89,187,110,180,103,173,96,194)(90,172,111,193,104,186,97,179)(91,185,112,178,105,171,98,192)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112)(113,146)(114,147)(115,148)(116,149)(117,150)(118,151)(119,152)(120,153)(121,154)(122,155)(123,156)(124,157)(125,158)(126,159)(127,160)(128,161)(129,162)(130,163)(131,164)(132,165)(133,166)(134,167)(135,168)(136,141)(137,142)(138,143)(139,144)(140,145)(169,215)(170,216)(171,217)(172,218)(173,219)(174,220)(175,221)(176,222)(177,223)(178,224)(179,197)(180,198)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)(188,206)(189,207)(190,208)(191,209)(192,210)(193,211)(194,212)(195,213)(196,214), (1,22,15,8)(2,23,16,9)(3,24,17,10)(4,25,18,11)(5,26,19,12)(6,27,20,13)(7,28,21,14)(29,50,43,36)(30,51,44,37)(31,52,45,38)(32,53,46,39)(33,54,47,40)(34,55,48,41)(35,56,49,42)(57,64,71,78)(58,65,72,79)(59,66,73,80)(60,67,74,81)(61,68,75,82)(62,69,76,83)(63,70,77,84)(85,92,99,106)(86,93,100,107)(87,94,101,108)(88,95,102,109)(89,96,103,110)(90,97,104,111)(91,98,105,112)(113,134,127,120)(114,135,128,121)(115,136,129,122)(116,137,130,123)(117,138,131,124)(118,139,132,125)(119,140,133,126)(141,162,155,148)(142,163,156,149)(143,164,157,150)(144,165,158,151)(145,166,159,152)(146,167,160,153)(147,168,161,154)(169,176,183,190)(170,177,184,191)(171,178,185,192)(172,179,186,193)(173,180,187,194)(174,181,188,195)(175,182,189,196)(197,204,211,218)(198,205,212,219)(199,206,213,220)(200,207,214,221)(201,208,215,222)(202,209,216,223)(203,210,217,224), (1,57)(2,58)(3,59)(4,60)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(127,169)(128,170)(129,171)(130,172)(131,173)(132,174)(133,175)(134,176)(135,177)(136,178)(137,179)(138,180)(139,181)(140,182)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,154,22,147,15,168,8,161)(2,167,23,160,16,153,9,146)(3,152,24,145,17,166,10,159)(4,165,25,158,18,151,11,144)(5,150,26,143,19,164,12,157)(6,163,27,156,20,149,13,142)(7,148,28,141,21,162,14,155)(29,121,50,114,43,135,36,128)(30,134,51,127,44,120,37,113)(31,119,52,140,45,133,38,126)(32,132,53,125,46,118,39,139)(33,117,54,138,47,131,40,124)(34,130,55,123,48,116,41,137)(35,115,56,136,49,129,42,122)(57,209,78,202,71,223,64,216)(58,222,79,215,72,208,65,201)(59,207,80,200,73,221,66,214)(60,220,81,213,74,206,67,199)(61,205,82,198,75,219,68,212)(62,218,83,211,76,204,69,197)(63,203,84,224,77,217,70,210)(85,191,106,184,99,177,92,170)(86,176,107,169,100,190,93,183)(87,189,108,182,101,175,94,196)(88,174,109,195,102,188,95,181)(89,187,110,180,103,173,96,194)(90,172,111,193,104,186,97,179)(91,185,112,178,105,171,98,192) );
G=PermutationGroup([(1,29),(2,30),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112),(113,146),(114,147),(115,148),(116,149),(117,150),(118,151),(119,152),(120,153),(121,154),(122,155),(123,156),(124,157),(125,158),(126,159),(127,160),(128,161),(129,162),(130,163),(131,164),(132,165),(133,166),(134,167),(135,168),(136,141),(137,142),(138,143),(139,144),(140,145),(169,215),(170,216),(171,217),(172,218),(173,219),(174,220),(175,221),(176,222),(177,223),(178,224),(179,197),(180,198),(181,199),(182,200),(183,201),(184,202),(185,203),(186,204),(187,205),(188,206),(189,207),(190,208),(191,209),(192,210),(193,211),(194,212),(195,213),(196,214)], [(1,22,15,8),(2,23,16,9),(3,24,17,10),(4,25,18,11),(5,26,19,12),(6,27,20,13),(7,28,21,14),(29,50,43,36),(30,51,44,37),(31,52,45,38),(32,53,46,39),(33,54,47,40),(34,55,48,41),(35,56,49,42),(57,64,71,78),(58,65,72,79),(59,66,73,80),(60,67,74,81),(61,68,75,82),(62,69,76,83),(63,70,77,84),(85,92,99,106),(86,93,100,107),(87,94,101,108),(88,95,102,109),(89,96,103,110),(90,97,104,111),(91,98,105,112),(113,134,127,120),(114,135,128,121),(115,136,129,122),(116,137,130,123),(117,138,131,124),(118,139,132,125),(119,140,133,126),(141,162,155,148),(142,163,156,149),(143,164,157,150),(144,165,158,151),(145,166,159,152),(146,167,160,153),(147,168,161,154),(169,176,183,190),(170,177,184,191),(171,178,185,192),(172,179,186,193),(173,180,187,194),(174,181,188,195),(175,182,189,196),(197,204,211,218),(198,205,212,219),(199,206,213,220),(200,207,214,221),(201,208,215,222),(202,209,216,223),(203,210,217,224)], [(1,57),(2,58),(3,59),(4,60),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112),(113,183),(114,184),(115,185),(116,186),(117,187),(118,188),(119,189),(120,190),(121,191),(122,192),(123,193),(124,194),(125,195),(126,196),(127,169),(128,170),(129,171),(130,172),(131,173),(132,174),(133,175),(134,176),(135,177),(136,178),(137,179),(138,180),(139,181),(140,182),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(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)], [(1,154,22,147,15,168,8,161),(2,167,23,160,16,153,9,146),(3,152,24,145,17,166,10,159),(4,165,25,158,18,151,11,144),(5,150,26,143,19,164,12,157),(6,163,27,156,20,149,13,142),(7,148,28,141,21,162,14,155),(29,121,50,114,43,135,36,128),(30,134,51,127,44,120,37,113),(31,119,52,140,45,133,38,126),(32,132,53,125,46,118,39,139),(33,117,54,138,47,131,40,124),(34,130,55,123,48,116,41,137),(35,115,56,136,49,129,42,122),(57,209,78,202,71,223,64,216),(58,222,79,215,72,208,65,201),(59,207,80,200,73,221,66,214),(60,220,81,213,74,206,67,199),(61,205,82,198,75,219,68,212),(62,218,83,211,76,204,69,197),(63,203,84,224,77,217,70,210),(85,191,106,184,99,177,92,170),(86,176,107,169,100,190,93,183),(87,189,108,182,101,175,94,196),(88,174,109,195,102,188,95,181),(89,187,110,180,103,173,96,194),(90,172,111,193,104,186,97,179),(91,185,112,178,105,171,98,192)])
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | 7B | 7C | 8A | ··· | 8H | 8I | ··· | 8T | 14A | ··· | 14I | 14J | ··· | 14AA | 28A | ··· | 28L | 28M | ··· | 28AD |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 7 | ··· | 7 | 14 | ··· | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | - | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D7 | D14 | Dic7 | Dic7 | Dic7 | D14 | C8○D4 | D4.Dic7 |
| kernel | C2×D4.Dic7 | C22×C7⋊C8 | C2×C4.Dic7 | D4.Dic7 | C14×C4○D4 | D4×C14 | Q8×C14 | C7×C4○D4 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C14 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 6 | 2 | 8 | 3 | 9 | 9 | 3 | 12 | 12 | 8 | 12 |
Matrix representation of C2×D4.Dic7 ►in GL4(𝔽113) generated by
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 8 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 98 | 85 |
| 0 | 0 | 8 | 15 |
| 34 | 112 | 0 | 0 |
| 60 | 88 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
| 74 | 68 | 0 | 0 |
| 84 | 39 | 0 | 0 |
| 0 | 0 | 95 | 0 |
| 0 | 0 | 0 | 95 |
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,98,8,0,0,0,15],[1,0,0,0,0,1,0,0,0,0,98,8,0,0,85,15],[34,60,0,0,112,88,0,0,0,0,98,0,0,0,0,98],[74,84,0,0,68,39,0,0,0,0,95,0,0,0,0,95] >;
C2×D4.Dic7 in GAP, Magma, Sage, TeX
C_2\times D_4.{\rm Dic}_7 % in TeX
G:=Group("C2xD4.Dic7"); // GroupNames label
G:=SmallGroup(448,1271);
// by ID
G=gap.SmallGroup(448,1271);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,297,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=b^2,e^2=b^2*d^7,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,c*e=e*c,e*d*e^-1=d^13>;
// generators/relations