direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×C4.Q8, D14.7SD16, (C8×D7)⋊3C4, C8⋊14(C4×D7), C56⋊14(C2×C4), (C4×D7).3Q8, C4.23(Q8×D7), C8⋊Dic7⋊24C2, C28.12(C2×Q8), C2.6(D7×SD16), C4⋊C4.160D14, (C2×C8).258D14, C22.83(D4×D7), D14.10(C4⋊C4), Dic7.4(C4⋊C4), C28.42(C22×C4), C4.Dic14⋊14C2, (C2×Dic7).98D4, C14.37(C2×SD16), (C2×C56).159C22, (C2×C28).275C23, (C22×D7).106D4, C4⋊Dic7.107C22, C7⋊C8⋊21(C2×C4), C7⋊1(C2×C4.Q8), (D7×C2×C8).7C2, C4.76(C2×C4×D7), (D7×C4⋊C4).4C2, C2.11(D7×C4⋊C4), (C7×C4.Q8)⋊7C2, C14.10(C2×C4⋊C4), (C4×D7).25(C2×C4), (C2×C14).280(C2×D4), (C7×C4⋊C4).68C22, (C2×C7⋊C8).226C22, (C2×C4×D7).231C22, (C2×C4).378(C22×D7), SmallGroup(448,393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×C4.Q8
G = < a,b,c,d,e | a7=b2=c4=1, d4=c2, e2=c-1d2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >
Subgroups: 620 in 130 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, C23, D7, C14, C14, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, Dic7, Dic7, C28, C28, D14, C2×C14, C4.Q8, C4.Q8, C2×C4⋊C4, C22×C8, C7⋊C8, C56, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C2×C4.Q8, C8×D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×C4×D7, C2×C4×D7, C4.Dic14, C8⋊Dic7, C7×C4.Q8, D7×C4⋊C4, D7×C2×C8, D7×C4.Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D7, C4⋊C4, SD16, C22×C4, C2×D4, C2×Q8, D14, C4.Q8, C2×C4⋊C4, C2×SD16, C4×D7, C22×D7, C2×C4.Q8, C2×C4×D7, D4×D7, Q8×D7, D7×C4⋊C4, D7×SD16, D7×C4.Q8
►On 224 points
Generators in S224
(1 138 45 27 36 199 83)(2 139 46 28 37 200 84)(3 140 47 29 38 193 85)(4 141 48 30 39 194 86)(5 142 41 31 40 195 87)(6 143 42 32 33 196 88)(7 144 43 25 34 197 81)(8 137 44 26 35 198 82)(9 222 108 168 169 74 190)(10 223 109 161 170 75 191)(11 224 110 162 171 76 192)(12 217 111 163 172 77 185)(13 218 112 164 173 78 186)(14 219 105 165 174 79 187)(15 220 106 166 175 80 188)(16 221 107 167 176 73 189)(17 113 124 64 52 184 133)(18 114 125 57 53 177 134)(19 115 126 58 54 178 135)(20 116 127 59 55 179 136)(21 117 128 60 56 180 129)(22 118 121 61 49 181 130)(23 119 122 62 50 182 131)(24 120 123 63 51 183 132)(65 202 94 215 99 148 155)(66 203 95 216 100 149 156)(67 204 96 209 101 150 157)(68 205 89 210 102 151 158)(69 206 90 211 103 152 159)(70 207 91 212 104 145 160)(71 208 92 213 97 146 153)(72 201 93 214 98 147 154)
(1 83)(2 84)(3 85)(4 86)(5 87)(6 88)(7 81)(8 82)(9 168)(10 161)(11 162)(12 163)(13 164)(14 165)(15 166)(16 167)(17 124)(18 125)(19 126)(20 127)(21 128)(22 121)(23 122)(24 123)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)(49 181)(50 182)(51 183)(52 184)(53 177)(54 178)(55 179)(56 180)(57 134)(58 135)(59 136)(60 129)(61 130)(62 131)(63 132)(64 133)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(71 153)(72 154)(89 102)(90 103)(91 104)(92 97)(93 98)(94 99)(95 100)(96 101)(105 219)(106 220)(107 221)(108 222)(109 223)(110 224)(111 217)(112 218)(137 198)(138 199)(139 200)(140 193)(141 194)(142 195)(143 196)(144 197)(145 207)(146 208)(147 201)(148 202)(149 203)(150 204)(151 205)(152 206)(169 190)(170 191)(171 192)(172 185)(173 186)(174 187)(175 188)(176 189)
(1 67 5 71)(2 68 6 72)(3 69 7 65)(4 70 8 66)(9 61 13 57)(10 62 14 58)(11 63 15 59)(12 64 16 60)(17 176 21 172)(18 169 22 173)(19 170 23 174)(20 171 24 175)(25 215 29 211)(26 216 30 212)(27 209 31 213)(28 210 32 214)(33 98 37 102)(34 99 38 103)(35 100 39 104)(36 101 40 97)(41 92 45 96)(42 93 46 89)(43 94 47 90)(44 95 48 91)(49 218 53 222)(50 219 54 223)(51 220 55 224)(52 221 56 217)(73 117 77 113)(74 118 78 114)(75 119 79 115)(76 120 80 116)(81 155 85 159)(82 156 86 160)(83 157 87 153)(84 158 88 154)(105 178 109 182)(106 179 110 183)(107 180 111 184)(108 181 112 177)(121 186 125 190)(122 187 126 191)(123 188 127 192)(124 189 128 185)(129 163 133 167)(130 164 134 168)(131 165 135 161)(132 166 136 162)(137 203 141 207)(138 204 142 208)(139 205 143 201)(140 206 144 202)(145 198 149 194)(146 199 150 195)(147 200 151 196)(148 193 152 197)
(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)
(1 180 65 109)(2 183 66 112)(3 178 67 107)(4 181 68 110)(5 184 69 105)(6 179 70 108)(7 182 71 111)(8 177 72 106)(9 196 59 145)(10 199 60 148)(11 194 61 151)(12 197 62 146)(13 200 63 149)(14 195 64 152)(15 198 57 147)(16 193 58 150)(17 90 174 41)(18 93 175 44)(19 96 176 47)(20 91 169 42)(21 94 170 45)(22 89 171 48)(23 92 172 43)(24 95 173 46)(25 119 213 77)(26 114 214 80)(27 117 215 75)(28 120 216 78)(29 115 209 73)(30 118 210 76)(31 113 211 79)(32 116 212 74)(33 127 104 190)(34 122 97 185)(35 125 98 188)(36 128 99 191)(37 123 100 186)(38 126 101 189)(39 121 102 192)(40 124 103 187)(49 158 224 86)(50 153 217 81)(51 156 218 84)(52 159 219 87)(53 154 220 82)(54 157 221 85)(55 160 222 88)(56 155 223 83)(129 202 161 138)(130 205 162 141)(131 208 163 144)(132 203 164 139)(133 206 165 142)(134 201 166 137)(135 204 167 140)(136 207 168 143)
G:=sub<Sym(224)| (1,138,45,27,36,199,83)(2,139,46,28,37,200,84)(3,140,47,29,38,193,85)(4,141,48,30,39,194,86)(5,142,41,31,40,195,87)(6,143,42,32,33,196,88)(7,144,43,25,34,197,81)(8,137,44,26,35,198,82)(9,222,108,168,169,74,190)(10,223,109,161,170,75,191)(11,224,110,162,171,76,192)(12,217,111,163,172,77,185)(13,218,112,164,173,78,186)(14,219,105,165,174,79,187)(15,220,106,166,175,80,188)(16,221,107,167,176,73,189)(17,113,124,64,52,184,133)(18,114,125,57,53,177,134)(19,115,126,58,54,178,135)(20,116,127,59,55,179,136)(21,117,128,60,56,180,129)(22,118,121,61,49,181,130)(23,119,122,62,50,182,131)(24,120,123,63,51,183,132)(65,202,94,215,99,148,155)(66,203,95,216,100,149,156)(67,204,96,209,101,150,157)(68,205,89,210,102,151,158)(69,206,90,211,103,152,159)(70,207,91,212,104,145,160)(71,208,92,213,97,146,153)(72,201,93,214,98,147,154), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,168)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,124)(18,125)(19,126)(20,127)(21,128)(22,121)(23,122)(24,123)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(49,181)(50,182)(51,183)(52,184)(53,177)(54,178)(55,179)(56,180)(57,134)(58,135)(59,136)(60,129)(61,130)(62,131)(63,132)(64,133)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,153)(72,154)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101)(105,219)(106,220)(107,221)(108,222)(109,223)(110,224)(111,217)(112,218)(137,198)(138,199)(139,200)(140,193)(141,194)(142,195)(143,196)(144,197)(145,207)(146,208)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(169,190)(170,191)(171,192)(172,185)(173,186)(174,187)(175,188)(176,189), (1,67,5,71)(2,68,6,72)(3,69,7,65)(4,70,8,66)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,176,21,172)(18,169,22,173)(19,170,23,174)(20,171,24,175)(25,215,29,211)(26,216,30,212)(27,209,31,213)(28,210,32,214)(33,98,37,102)(34,99,38,103)(35,100,39,104)(36,101,40,97)(41,92,45,96)(42,93,46,89)(43,94,47,90)(44,95,48,91)(49,218,53,222)(50,219,54,223)(51,220,55,224)(52,221,56,217)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(81,155,85,159)(82,156,86,160)(83,157,87,153)(84,158,88,154)(105,178,109,182)(106,179,110,183)(107,180,111,184)(108,181,112,177)(121,186,125,190)(122,187,126,191)(123,188,127,192)(124,189,128,185)(129,163,133,167)(130,164,134,168)(131,165,135,161)(132,166,136,162)(137,203,141,207)(138,204,142,208)(139,205,143,201)(140,206,144,202)(145,198,149,194)(146,199,150,195)(147,200,151,196)(148,193,152,197), (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), (1,180,65,109)(2,183,66,112)(3,178,67,107)(4,181,68,110)(5,184,69,105)(6,179,70,108)(7,182,71,111)(8,177,72,106)(9,196,59,145)(10,199,60,148)(11,194,61,151)(12,197,62,146)(13,200,63,149)(14,195,64,152)(15,198,57,147)(16,193,58,150)(17,90,174,41)(18,93,175,44)(19,96,176,47)(20,91,169,42)(21,94,170,45)(22,89,171,48)(23,92,172,43)(24,95,173,46)(25,119,213,77)(26,114,214,80)(27,117,215,75)(28,120,216,78)(29,115,209,73)(30,118,210,76)(31,113,211,79)(32,116,212,74)(33,127,104,190)(34,122,97,185)(35,125,98,188)(36,128,99,191)(37,123,100,186)(38,126,101,189)(39,121,102,192)(40,124,103,187)(49,158,224,86)(50,153,217,81)(51,156,218,84)(52,159,219,87)(53,154,220,82)(54,157,221,85)(55,160,222,88)(56,155,223,83)(129,202,161,138)(130,205,162,141)(131,208,163,144)(132,203,164,139)(133,206,165,142)(134,201,166,137)(135,204,167,140)(136,207,168,143)>;
G:=Group( (1,138,45,27,36,199,83)(2,139,46,28,37,200,84)(3,140,47,29,38,193,85)(4,141,48,30,39,194,86)(5,142,41,31,40,195,87)(6,143,42,32,33,196,88)(7,144,43,25,34,197,81)(8,137,44,26,35,198,82)(9,222,108,168,169,74,190)(10,223,109,161,170,75,191)(11,224,110,162,171,76,192)(12,217,111,163,172,77,185)(13,218,112,164,173,78,186)(14,219,105,165,174,79,187)(15,220,106,166,175,80,188)(16,221,107,167,176,73,189)(17,113,124,64,52,184,133)(18,114,125,57,53,177,134)(19,115,126,58,54,178,135)(20,116,127,59,55,179,136)(21,117,128,60,56,180,129)(22,118,121,61,49,181,130)(23,119,122,62,50,182,131)(24,120,123,63,51,183,132)(65,202,94,215,99,148,155)(66,203,95,216,100,149,156)(67,204,96,209,101,150,157)(68,205,89,210,102,151,158)(69,206,90,211,103,152,159)(70,207,91,212,104,145,160)(71,208,92,213,97,146,153)(72,201,93,214,98,147,154), (1,83)(2,84)(3,85)(4,86)(5,87)(6,88)(7,81)(8,82)(9,168)(10,161)(11,162)(12,163)(13,164)(14,165)(15,166)(16,167)(17,124)(18,125)(19,126)(20,127)(21,128)(22,121)(23,122)(24,123)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41)(49,181)(50,182)(51,183)(52,184)(53,177)(54,178)(55,179)(56,180)(57,134)(58,135)(59,136)(60,129)(61,130)(62,131)(63,132)(64,133)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(71,153)(72,154)(89,102)(90,103)(91,104)(92,97)(93,98)(94,99)(95,100)(96,101)(105,219)(106,220)(107,221)(108,222)(109,223)(110,224)(111,217)(112,218)(137,198)(138,199)(139,200)(140,193)(141,194)(142,195)(143,196)(144,197)(145,207)(146,208)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(169,190)(170,191)(171,192)(172,185)(173,186)(174,187)(175,188)(176,189), (1,67,5,71)(2,68,6,72)(3,69,7,65)(4,70,8,66)(9,61,13,57)(10,62,14,58)(11,63,15,59)(12,64,16,60)(17,176,21,172)(18,169,22,173)(19,170,23,174)(20,171,24,175)(25,215,29,211)(26,216,30,212)(27,209,31,213)(28,210,32,214)(33,98,37,102)(34,99,38,103)(35,100,39,104)(36,101,40,97)(41,92,45,96)(42,93,46,89)(43,94,47,90)(44,95,48,91)(49,218,53,222)(50,219,54,223)(51,220,55,224)(52,221,56,217)(73,117,77,113)(74,118,78,114)(75,119,79,115)(76,120,80,116)(81,155,85,159)(82,156,86,160)(83,157,87,153)(84,158,88,154)(105,178,109,182)(106,179,110,183)(107,180,111,184)(108,181,112,177)(121,186,125,190)(122,187,126,191)(123,188,127,192)(124,189,128,185)(129,163,133,167)(130,164,134,168)(131,165,135,161)(132,166,136,162)(137,203,141,207)(138,204,142,208)(139,205,143,201)(140,206,144,202)(145,198,149,194)(146,199,150,195)(147,200,151,196)(148,193,152,197), (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), (1,180,65,109)(2,183,66,112)(3,178,67,107)(4,181,68,110)(5,184,69,105)(6,179,70,108)(7,182,71,111)(8,177,72,106)(9,196,59,145)(10,199,60,148)(11,194,61,151)(12,197,62,146)(13,200,63,149)(14,195,64,152)(15,198,57,147)(16,193,58,150)(17,90,174,41)(18,93,175,44)(19,96,176,47)(20,91,169,42)(21,94,170,45)(22,89,171,48)(23,92,172,43)(24,95,173,46)(25,119,213,77)(26,114,214,80)(27,117,215,75)(28,120,216,78)(29,115,209,73)(30,118,210,76)(31,113,211,79)(32,116,212,74)(33,127,104,190)(34,122,97,185)(35,125,98,188)(36,128,99,191)(37,123,100,186)(38,126,101,189)(39,121,102,192)(40,124,103,187)(49,158,224,86)(50,153,217,81)(51,156,218,84)(52,159,219,87)(53,154,220,82)(54,157,221,85)(55,160,222,88)(56,155,223,83)(129,202,161,138)(130,205,162,141)(131,208,163,144)(132,203,164,139)(133,206,165,142)(134,201,166,137)(135,204,167,140)(136,207,168,143) );
G=PermutationGroup([[(1,138,45,27,36,199,83),(2,139,46,28,37,200,84),(3,140,47,29,38,193,85),(4,141,48,30,39,194,86),(5,142,41,31,40,195,87),(6,143,42,32,33,196,88),(7,144,43,25,34,197,81),(8,137,44,26,35,198,82),(9,222,108,168,169,74,190),(10,223,109,161,170,75,191),(11,224,110,162,171,76,192),(12,217,111,163,172,77,185),(13,218,112,164,173,78,186),(14,219,105,165,174,79,187),(15,220,106,166,175,80,188),(16,221,107,167,176,73,189),(17,113,124,64,52,184,133),(18,114,125,57,53,177,134),(19,115,126,58,54,178,135),(20,116,127,59,55,179,136),(21,117,128,60,56,180,129),(22,118,121,61,49,181,130),(23,119,122,62,50,182,131),(24,120,123,63,51,183,132),(65,202,94,215,99,148,155),(66,203,95,216,100,149,156),(67,204,96,209,101,150,157),(68,205,89,210,102,151,158),(69,206,90,211,103,152,159),(70,207,91,212,104,145,160),(71,208,92,213,97,146,153),(72,201,93,214,98,147,154)], [(1,83),(2,84),(3,85),(4,86),(5,87),(6,88),(7,81),(8,82),(9,168),(10,161),(11,162),(12,163),(13,164),(14,165),(15,166),(16,167),(17,124),(18,125),(19,126),(20,127),(21,128),(22,121),(23,122),(24,123),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41),(49,181),(50,182),(51,183),(52,184),(53,177),(54,178),(55,179),(56,180),(57,134),(58,135),(59,136),(60,129),(61,130),(62,131),(63,132),(64,133),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(71,153),(72,154),(89,102),(90,103),(91,104),(92,97),(93,98),(94,99),(95,100),(96,101),(105,219),(106,220),(107,221),(108,222),(109,223),(110,224),(111,217),(112,218),(137,198),(138,199),(139,200),(140,193),(141,194),(142,195),(143,196),(144,197),(145,207),(146,208),(147,201),(148,202),(149,203),(150,204),(151,205),(152,206),(169,190),(170,191),(171,192),(172,185),(173,186),(174,187),(175,188),(176,189)], [(1,67,5,71),(2,68,6,72),(3,69,7,65),(4,70,8,66),(9,61,13,57),(10,62,14,58),(11,63,15,59),(12,64,16,60),(17,176,21,172),(18,169,22,173),(19,170,23,174),(20,171,24,175),(25,215,29,211),(26,216,30,212),(27,209,31,213),(28,210,32,214),(33,98,37,102),(34,99,38,103),(35,100,39,104),(36,101,40,97),(41,92,45,96),(42,93,46,89),(43,94,47,90),(44,95,48,91),(49,218,53,222),(50,219,54,223),(51,220,55,224),(52,221,56,217),(73,117,77,113),(74,118,78,114),(75,119,79,115),(76,120,80,116),(81,155,85,159),(82,156,86,160),(83,157,87,153),(84,158,88,154),(105,178,109,182),(106,179,110,183),(107,180,111,184),(108,181,112,177),(121,186,125,190),(122,187,126,191),(123,188,127,192),(124,189,128,185),(129,163,133,167),(130,164,134,168),(131,165,135,161),(132,166,136,162),(137,203,141,207),(138,204,142,208),(139,205,143,201),(140,206,144,202),(145,198,149,194),(146,199,150,195),(147,200,151,196),(148,193,152,197)], [(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)], [(1,180,65,109),(2,183,66,112),(3,178,67,107),(4,181,68,110),(5,184,69,105),(6,179,70,108),(7,182,71,111),(8,177,72,106),(9,196,59,145),(10,199,60,148),(11,194,61,151),(12,197,62,146),(13,200,63,149),(14,195,64,152),(15,198,57,147),(16,193,58,150),(17,90,174,41),(18,93,175,44),(19,96,176,47),(20,91,169,42),(21,94,170,45),(22,89,171,48),(23,92,172,43),(24,95,173,46),(25,119,213,77),(26,114,214,80),(27,117,215,75),(28,120,216,78),(29,115,209,73),(30,118,210,76),(31,113,211,79),(32,116,212,74),(33,127,104,190),(34,122,97,185),(35,125,98,188),(36,128,99,191),(37,123,100,186),(38,126,101,189),(39,121,102,192),(40,124,103,187),(49,158,224,86),(50,153,217,81),(51,156,218,84),(52,159,219,87),(53,154,220,82),(54,157,221,85),(55,160,222,88),(56,155,223,83),(129,202,161,138),(130,205,162,141),(131,208,163,144),(132,203,164,139),(133,206,165,142),(134,201,166,137),(135,204,167,140),(136,207,168,143)]])
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 14A | ··· | 14I | 28A | ··· | 28F | 28G | ··· | 28R | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | 2 | 4 | 4 | 4 | 4 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 | 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 | C4 | Q8 | D4 | D4 | D7 | SD16 | D14 | D14 | C4×D7 | Q8×D7 | D4×D7 | D7×SD16 |
| kernel | D7×C4.Q8 | C4.Dic14 | C8⋊Dic7 | C7×C4.Q8 | D7×C4⋊C4 | D7×C2×C8 | C8×D7 | C4×D7 | C2×Dic7 | C22×D7 | C4.Q8 | D14 | C4⋊C4 | C2×C8 | C8 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 2 | 1 | 1 | 3 | 8 | 6 | 3 | 12 | 3 | 3 | 12 |
Matrix representation of D7×C4.Q8 ►in GL4(𝔽113) generated by
| 0 | 1 | 0 | 0 |
| 112 | 24 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 13 | 100 |
| 0 | 0 | 13 | 13 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 11 | 85 |
| 0 | 0 | 85 | 102 |
G:=sub<GL(4,GF(113))| [0,112,0,0,1,24,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,0,112,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,13,0,0,100,13],[98,0,0,0,0,98,0,0,0,0,11,85,0,0,85,102] >;
D7×C4.Q8 in GAP, Magma, Sage, TeX
D_7\times C_4.Q_8
% in TeX
G:=Group("D7xC4.Q8"); // GroupNames label
G:=SmallGroup(448,393);
// by ID
G=gap.SmallGroup(448,393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,555,58,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=1,d^4=c^2,e^2=c^-1*d^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^3>;
// generators/relations