direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.Dic7, C28.75C24, C14⋊3(C8○D4), C7⋊C8.37C23, C4○D4.44D14, (D4×C14).12C4, (Q8×C14).12C4, C4○D4.4Dic7, Q8.9(C2×Dic7), D4.8(C2×Dic7), C4.74(C23×D7), C28.97(C22×C4), C14.48(C23×C4), (C2×Q8).10Dic7, (C2×D4).12Dic7, (C2×C28).553C23, (C22×C4).386D14, C4.Dic7⋊34C22, C23.19(C2×Dic7), C4.40(C22×Dic7), C2.10(C23×Dic7), (C22×C28).288C22, C22.33(C22×Dic7), C7⋊4(C2×C8○D4), (C22×C7⋊C8)⋊14C2, (C2×C7⋊C8)⋊41C22, (C7×C4○D4).3C4, (C2×C4○D4).13D7, (C14×C4○D4).9C2, (C7×D4).25(C2×C4), (C7×Q8).27(C2×C4), (C2×C28).135(C2×C4), (C2×C4.Dic7)⋊28C2, (C2×C4).56(C2×Dic7), (C2×C14).28(C22×C4), (C22×C14).80(C2×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×Q8.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, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C7⋊C8, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C2×C8○D4, C2×C7⋊C8, C2×C7⋊C8, C4.Dic7, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C22×C7⋊C8, C2×C4.Dic7, Q8.Dic7, C14×C4○D4, C2×Q8.Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C24, Dic7, D14, C8○D4, C23×C4, C2×Dic7, C22×D7, C2×C8○D4, C22×Dic7, C23×D7, Q8.Dic7, C23×Dic7, C2×Q8.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)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 141)(122 142)(123 143)(124 144)(125 145)(126 146)(127 147)(128 148)(129 149)(130 150)(131 151)(132 152)(133 153)(134 154)(135 155)(136 156)(137 157)(138 158)(139 159)(140 160)(169 223)(170 224)(171 197)(172 198)(173 199)(174 200)(175 201)(176 202)(177 203)(178 204)(179 205)(180 206)(181 207)(182 208)(183 209)(184 210)(185 211)(186 212)(187 213)(188 214)(189 215)(190 216)(191 217)(192 218)(193 219)(194 220)(195 221)(196 222)
(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 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 73)(11 74)(12 75)(13 76)(14 77)(15 78)(16 79)(17 80)(18 81)(19 82)(20 83)(21 84)(22 57)(23 58)(24 59)(25 60)(26 61)(27 62)(28 63)(29 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 99)(37 100)(38 101)(39 102)(40 103)(41 104)(42 105)(43 106)(44 107)(45 108)(46 109)(47 110)(48 111)(49 112)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(113 190)(114 191)(115 192)(116 193)(117 194)(118 195)(119 196)(120 169)(121 170)(122 171)(123 172)(124 173)(125 174)(126 175)(127 176)(128 177)(129 178)(130 179)(131 180)(132 181)(133 182)(134 183)(135 184)(136 185)(137 186)(138 187)(139 188)(140 189)(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)(29 134 50 127 43 120 36 113)(30 119 51 140 44 133 37 126)(31 132 52 125 45 118 38 139)(32 117 53 138 46 131 39 124)(33 130 54 123 47 116 40 137)(34 115 55 136 48 129 41 122)(35 128 56 121 49 114 42 135)(57 202 78 223 71 216 64 209)(58 215 79 208 72 201 65 222)(59 200 80 221 73 214 66 207)(60 213 81 206 74 199 67 220)(61 198 82 219 75 212 68 205)(62 211 83 204 76 197 69 218)(63 224 84 217 77 210 70 203)(85 176 106 169 99 190 92 183)(86 189 107 182 100 175 93 196)(87 174 108 195 101 188 94 181)(88 187 109 180 102 173 95 194)(89 172 110 193 103 186 96 179)(90 185 111 178 104 171 97 192)(91 170 112 191 105 184 98 177)
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)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(169,223)(170,224)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,209)(184,210)(185,211)(186,212)(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221)(196,222), (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), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(113,190)(114,191)(115,192)(116,193)(117,194)(118,195)(119,196)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,181)(133,182)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(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)(29,134,50,127,43,120,36,113)(30,119,51,140,44,133,37,126)(31,132,52,125,45,118,38,139)(32,117,53,138,46,131,39,124)(33,130,54,123,47,116,40,137)(34,115,55,136,48,129,41,122)(35,128,56,121,49,114,42,135)(57,202,78,223,71,216,64,209)(58,215,79,208,72,201,65,222)(59,200,80,221,73,214,66,207)(60,213,81,206,74,199,67,220)(61,198,82,219,75,212,68,205)(62,211,83,204,76,197,69,218)(63,224,84,217,77,210,70,203)(85,176,106,169,99,190,92,183)(86,189,107,182,100,175,93,196)(87,174,108,195,101,188,94,181)(88,187,109,180,102,173,95,194)(89,172,110,193,103,186,96,179)(90,185,111,178,104,171,97,192)(91,170,112,191,105,184,98,177)>;
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)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160)(169,223)(170,224)(171,197)(172,198)(173,199)(174,200)(175,201)(176,202)(177,203)(178,204)(179,205)(180,206)(181,207)(182,208)(183,209)(184,210)(185,211)(186,212)(187,213)(188,214)(189,215)(190,216)(191,217)(192,218)(193,219)(194,220)(195,221)(196,222), (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), (1,64)(2,65)(3,66)(4,67)(5,68)(6,69)(7,70)(8,71)(9,72)(10,73)(11,74)(12,75)(13,76)(14,77)(15,78)(16,79)(17,80)(18,81)(19,82)(20,83)(21,84)(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,92)(30,93)(31,94)(32,95)(33,96)(34,97)(35,98)(36,99)(37,100)(38,101)(39,102)(40,103)(41,104)(42,105)(43,106)(44,107)(45,108)(46,109)(47,110)(48,111)(49,112)(50,85)(51,86)(52,87)(53,88)(54,89)(55,90)(56,91)(113,190)(114,191)(115,192)(116,193)(117,194)(118,195)(119,196)(120,169)(121,170)(122,171)(123,172)(124,173)(125,174)(126,175)(127,176)(128,177)(129,178)(130,179)(131,180)(132,181)(133,182)(134,183)(135,184)(136,185)(137,186)(138,187)(139,188)(140,189)(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)(29,134,50,127,43,120,36,113)(30,119,51,140,44,133,37,126)(31,132,52,125,45,118,38,139)(32,117,53,138,46,131,39,124)(33,130,54,123,47,116,40,137)(34,115,55,136,48,129,41,122)(35,128,56,121,49,114,42,135)(57,202,78,223,71,216,64,209)(58,215,79,208,72,201,65,222)(59,200,80,221,73,214,66,207)(60,213,81,206,74,199,67,220)(61,198,82,219,75,212,68,205)(62,211,83,204,76,197,69,218)(63,224,84,217,77,210,70,203)(85,176,106,169,99,190,92,183)(86,189,107,182,100,175,93,196)(87,174,108,195,101,188,94,181)(88,187,109,180,102,173,95,194)(89,172,110,193,103,186,96,179)(90,185,111,178,104,171,97,192)(91,170,112,191,105,184,98,177) );
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),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,141),(122,142),(123,143),(124,144),(125,145),(126,146),(127,147),(128,148),(129,149),(130,150),(131,151),(132,152),(133,153),(134,154),(135,155),(136,156),(137,157),(138,158),(139,159),(140,160),(169,223),(170,224),(171,197),(172,198),(173,199),(174,200),(175,201),(176,202),(177,203),(178,204),(179,205),(180,206),(181,207),(182,208),(183,209),(184,210),(185,211),(186,212),(187,213),(188,214),(189,215),(190,216),(191,217),(192,218),(193,219),(194,220),(195,221),(196,222)], [(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)], [(1,64),(2,65),(3,66),(4,67),(5,68),(6,69),(7,70),(8,71),(9,72),(10,73),(11,74),(12,75),(13,76),(14,77),(15,78),(16,79),(17,80),(18,81),(19,82),(20,83),(21,84),(22,57),(23,58),(24,59),(25,60),(26,61),(27,62),(28,63),(29,92),(30,93),(31,94),(32,95),(33,96),(34,97),(35,98),(36,99),(37,100),(38,101),(39,102),(40,103),(41,104),(42,105),(43,106),(44,107),(45,108),(46,109),(47,110),(48,111),(49,112),(50,85),(51,86),(52,87),(53,88),(54,89),(55,90),(56,91),(113,190),(114,191),(115,192),(116,193),(117,194),(118,195),(119,196),(120,169),(121,170),(122,171),(123,172),(124,173),(125,174),(126,175),(127,176),(128,177),(129,178),(130,179),(131,180),(132,181),(133,182),(134,183),(135,184),(136,185),(137,186),(138,187),(139,188),(140,189),(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),(29,134,50,127,43,120,36,113),(30,119,51,140,44,133,37,126),(31,132,52,125,45,118,38,139),(32,117,53,138,46,131,39,124),(33,130,54,123,47,116,40,137),(34,115,55,136,48,129,41,122),(35,128,56,121,49,114,42,135),(57,202,78,223,71,216,64,209),(58,215,79,208,72,201,65,222),(59,200,80,221,73,214,66,207),(60,213,81,206,74,199,67,220),(61,198,82,219,75,212,68,205),(62,211,83,204,76,197,69,218),(63,224,84,217,77,210,70,203),(85,176,106,169,99,190,92,183),(86,189,107,182,100,175,93,196),(87,174,108,195,101,188,94,181),(88,187,109,180,102,173,95,194),(89,172,110,193,103,186,96,179),(90,185,111,178,104,171,97,192),(91,170,112,191,105,184,98,177)]])
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 | Q8.Dic7 |
| kernel | C2×Q8.Dic7 | C22×C7⋊C8 | C2×C4.Dic7 | Q8.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×Q8.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×Q8.Dic7 in GAP, Magma, Sage, TeX
C_2\times Q_8.{\rm Dic}_7 % in TeX
G:=Group("C2xQ8.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