metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.D7⋊1C4, D4.1(C4×D7), C56⋊C4⋊15C2, C14.31(C4×D4), C4⋊C4.129D14, Dic14⋊1(C2×C4), D4⋊C4.7D7, (C2×C8).165D14, C28.Q8⋊2C2, C28.2(C22×C4), (D4×Dic7).1C2, C22.66(D4×D7), Dic7⋊3Q8⋊1C2, (C2×D4).124D14, C2.1(D8⋊D7), C7⋊2(SD16⋊C4), C28.143(C4○D4), C4.44(D4⋊2D7), C28.44D4⋊14C2, C14.24(C8⋊C22), (C2×C56).176C22, (C2×C28).197C23, (C2×Dic7).133D4, C2.1(SD16⋊D7), (D4×C14).18C22, C4⋊Dic7.57C22, (C4×Dic7).3C22, C14.24(C8.C22), C2.15(Dic7⋊4D4), (C2×Dic14).49C22, C7⋊C8⋊1(C2×C4), C4.2(C2×C4×D7), (C7×D4).1(C2×C4), (C2×C7⋊C8).5C22, (C7×C4⋊C4).2C22, (C2×D4.D7).1C2, (C7×D4⋊C4).7C2, (C2×C14).210(C2×D4), (C2×C4).304(C22×D7), SmallGroup(448,291)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D7⋊C4
G = < a,b,c,d,e | a4=b2=c7=e4=1, d2=a2, bab=dad-1=eae-1=a-1, ac=ca, bc=cb, dbd-1=ebe-1=ab, dcd-1=c-1, ce=ec, ede-1=a2d >
Subgroups: 532 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic7, C28, C28, C2×C14, C2×C14, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C7⋊C8, C56, Dic14, Dic14, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C22×C14, SD16⋊C4, C2×C7⋊C8, C4×Dic7, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D4.D7, C23.D7, C7×C4⋊C4, C2×C56, C2×Dic14, C22×Dic7, D4×C14, C28.Q8, C56⋊C4, C28.44D4, C7×D4⋊C4, Dic7⋊3Q8, C2×D4.D7, D4×Dic7, D4.D7⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C8⋊C22, C8.C22, C4×D7, C22×D7, SD16⋊C4, C2×C4×D7, D4×D7, D4⋊2D7, Dic7⋊4D4, D8⋊D7, SD16⋊D7, D4.D7⋊C4
►On 224 points
Generators in S224
(1 29 8 36)(2 30 9 37)(3 31 10 38)(4 32 11 39)(5 33 12 40)(6 34 13 41)(7 35 14 42)(15 43 22 50)(16 44 23 51)(17 45 24 52)(18 46 25 53)(19 47 26 54)(20 48 27 55)(21 49 28 56)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)(113 141 120 148)(114 142 121 149)(115 143 122 150)(116 144 123 151)(117 145 124 152)(118 146 125 153)(119 147 126 154)(127 155 134 162)(128 156 135 163)(129 157 136 164)(130 158 137 165)(131 159 138 166)(132 160 139 167)(133 161 140 168)(169 204 176 197)(170 205 177 198)(171 206 178 199)(172 207 179 200)(173 208 180 201)(174 209 181 202)(175 210 182 203)(183 218 190 211)(184 219 191 212)(185 220 192 213)(186 221 193 214)(187 222 194 215)(188 223 195 216)(189 224 196 217)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 29)(9 30)(10 31)(11 32)(12 33)(13 34)(14 35)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 141)(121 142)(122 143)(123 144)(124 145)(125 146)(126 147)(127 162)(128 163)(129 164)(130 165)(131 166)(132 167)(133 168)(134 155)(135 156)(136 157)(137 158)(138 159)(139 160)(140 161)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(183 190)(184 191)(185 192)(186 193)(187 194)(188 195)(189 196)
(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)
(1 194 8 187)(2 193 9 186)(3 192 10 185)(4 191 11 184)(5 190 12 183)(6 196 13 189)(7 195 14 188)(15 180 22 173)(16 179 23 172)(17 178 24 171)(18 177 25 170)(19 176 26 169)(20 182 27 175)(21 181 28 174)(29 222 36 215)(30 221 37 214)(31 220 38 213)(32 219 39 212)(33 218 40 211)(34 224 41 217)(35 223 42 216)(43 208 50 201)(44 207 51 200)(45 206 52 199)(46 205 53 198)(47 204 54 197)(48 210 55 203)(49 209 56 202)(57 124 64 117)(58 123 65 116)(59 122 66 115)(60 121 67 114)(61 120 68 113)(62 126 69 119)(63 125 70 118)(71 138 78 131)(72 137 79 130)(73 136 80 129)(74 135 81 128)(75 134 82 127)(76 140 83 133)(77 139 84 132)(85 152 92 145)(86 151 93 144)(87 150 94 143)(88 149 95 142)(89 148 96 141)(90 154 97 147)(91 153 98 146)(99 166 106 159)(100 165 107 158)(101 164 108 157)(102 163 109 156)(103 162 110 155)(104 168 111 161)(105 167 112 160)
(1 71 15 57)(2 72 16 58)(3 73 17 59)(4 74 18 60)(5 75 19 61)(6 76 20 62)(7 77 21 63)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)(113 190 127 176)(114 191 128 177)(115 192 129 178)(116 193 130 179)(117 194 131 180)(118 195 132 181)(119 196 133 182)(120 183 134 169)(121 184 135 170)(122 185 136 171)(123 186 137 172)(124 187 138 173)(125 188 139 174)(126 189 140 175)(141 218 155 204)(142 219 156 205)(143 220 157 206)(144 221 158 207)(145 222 159 208)(146 223 160 209)(147 224 161 210)(148 211 162 197)(149 212 163 198)(150 213 164 199)(151 214 165 200)(152 215 166 201)(153 216 167 202)(154 217 168 203)
G:=sub<Sym(224)| (1,29,8,36)(2,30,9,37)(3,31,10,38)(4,32,11,39)(5,33,12,40)(6,34,13,41)(7,35,14,42)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105)(113,141,120,148)(114,142,121,149)(115,143,122,150)(116,144,123,151)(117,145,124,152)(118,146,125,153)(119,147,126,154)(127,155,134,162)(128,156,135,163)(129,157,136,164)(130,158,137,165)(131,159,138,166)(132,160,139,167)(133,161,140,168)(169,204,176,197)(170,205,177,198)(171,206,178,199)(172,207,179,200)(173,208,180,201)(174,209,181,202)(175,210,182,203)(183,218,190,211)(184,219,191,212)(185,220,192,213)(186,221,193,214)(187,222,194,215)(188,223,195,216)(189,224,196,217), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196), (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), (1,194,8,187)(2,193,9,186)(3,192,10,185)(4,191,11,184)(5,190,12,183)(6,196,13,189)(7,195,14,188)(15,180,22,173)(16,179,23,172)(17,178,24,171)(18,177,25,170)(19,176,26,169)(20,182,27,175)(21,181,28,174)(29,222,36,215)(30,221,37,214)(31,220,38,213)(32,219,39,212)(33,218,40,211)(34,224,41,217)(35,223,42,216)(43,208,50,201)(44,207,51,200)(45,206,52,199)(46,205,53,198)(47,204,54,197)(48,210,55,203)(49,209,56,202)(57,124,64,117)(58,123,65,116)(59,122,66,115)(60,121,67,114)(61,120,68,113)(62,126,69,119)(63,125,70,118)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,140,83,133)(77,139,84,132)(85,152,92,145)(86,151,93,144)(87,150,94,143)(88,149,95,142)(89,148,96,141)(90,154,97,147)(91,153,98,146)(99,166,106,159)(100,165,107,158)(101,164,108,157)(102,163,109,156)(103,162,110,155)(104,168,111,161)(105,167,112,160), (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,190,127,176)(114,191,128,177)(115,192,129,178)(116,193,130,179)(117,194,131,180)(118,195,132,181)(119,196,133,182)(120,183,134,169)(121,184,135,170)(122,185,136,171)(123,186,137,172)(124,187,138,173)(125,188,139,174)(126,189,140,175)(141,218,155,204)(142,219,156,205)(143,220,157,206)(144,221,158,207)(145,222,159,208)(146,223,160,209)(147,224,161,210)(148,211,162,197)(149,212,163,198)(150,213,164,199)(151,214,165,200)(152,215,166,201)(153,216,167,202)(154,217,168,203)>;
G:=Group( (1,29,8,36)(2,30,9,37)(3,31,10,38)(4,32,11,39)(5,33,12,40)(6,34,13,41)(7,35,14,42)(15,43,22,50)(16,44,23,51)(17,45,24,52)(18,46,25,53)(19,47,26,54)(20,48,27,55)(21,49,28,56)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105)(113,141,120,148)(114,142,121,149)(115,143,122,150)(116,144,123,151)(117,145,124,152)(118,146,125,153)(119,147,126,154)(127,155,134,162)(128,156,135,163)(129,157,136,164)(130,158,137,165)(131,159,138,166)(132,160,139,167)(133,161,140,168)(169,204,176,197)(170,205,177,198)(171,206,178,199)(172,207,179,200)(173,208,180,201)(174,209,181,202)(175,210,182,203)(183,218,190,211)(184,219,191,212)(185,220,192,213)(186,221,193,214)(187,222,194,215)(188,223,195,216)(189,224,196,217), (1,36)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,29)(9,30)(10,31)(11,32)(12,33)(13,34)(14,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,141)(121,142)(122,143)(123,144)(124,145)(125,146)(126,147)(127,162)(128,163)(129,164)(130,165)(131,166)(132,167)(133,168)(134,155)(135,156)(136,157)(137,158)(138,159)(139,160)(140,161)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(183,190)(184,191)(185,192)(186,193)(187,194)(188,195)(189,196), (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), (1,194,8,187)(2,193,9,186)(3,192,10,185)(4,191,11,184)(5,190,12,183)(6,196,13,189)(7,195,14,188)(15,180,22,173)(16,179,23,172)(17,178,24,171)(18,177,25,170)(19,176,26,169)(20,182,27,175)(21,181,28,174)(29,222,36,215)(30,221,37,214)(31,220,38,213)(32,219,39,212)(33,218,40,211)(34,224,41,217)(35,223,42,216)(43,208,50,201)(44,207,51,200)(45,206,52,199)(46,205,53,198)(47,204,54,197)(48,210,55,203)(49,209,56,202)(57,124,64,117)(58,123,65,116)(59,122,66,115)(60,121,67,114)(61,120,68,113)(62,126,69,119)(63,125,70,118)(71,138,78,131)(72,137,79,130)(73,136,80,129)(74,135,81,128)(75,134,82,127)(76,140,83,133)(77,139,84,132)(85,152,92,145)(86,151,93,144)(87,150,94,143)(88,149,95,142)(89,148,96,141)(90,154,97,147)(91,153,98,146)(99,166,106,159)(100,165,107,158)(101,164,108,157)(102,163,109,156)(103,162,110,155)(104,168,111,161)(105,167,112,160), (1,71,15,57)(2,72,16,58)(3,73,17,59)(4,74,18,60)(5,75,19,61)(6,76,20,62)(7,77,21,63)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98)(113,190,127,176)(114,191,128,177)(115,192,129,178)(116,193,130,179)(117,194,131,180)(118,195,132,181)(119,196,133,182)(120,183,134,169)(121,184,135,170)(122,185,136,171)(123,186,137,172)(124,187,138,173)(125,188,139,174)(126,189,140,175)(141,218,155,204)(142,219,156,205)(143,220,157,206)(144,221,158,207)(145,222,159,208)(146,223,160,209)(147,224,161,210)(148,211,162,197)(149,212,163,198)(150,213,164,199)(151,214,165,200)(152,215,166,201)(153,216,167,202)(154,217,168,203) );
G=PermutationGroup([[(1,29,8,36),(2,30,9,37),(3,31,10,38),(4,32,11,39),(5,33,12,40),(6,34,13,41),(7,35,14,42),(15,43,22,50),(16,44,23,51),(17,45,24,52),(18,46,25,53),(19,47,26,54),(20,48,27,55),(21,49,28,56),(57,92,64,85),(58,93,65,86),(59,94,66,87),(60,95,67,88),(61,96,68,89),(62,97,69,90),(63,98,70,91),(71,106,78,99),(72,107,79,100),(73,108,80,101),(74,109,81,102),(75,110,82,103),(76,111,83,104),(77,112,84,105),(113,141,120,148),(114,142,121,149),(115,143,122,150),(116,144,123,151),(117,145,124,152),(118,146,125,153),(119,147,126,154),(127,155,134,162),(128,156,135,163),(129,157,136,164),(130,158,137,165),(131,159,138,166),(132,160,139,167),(133,161,140,168),(169,204,176,197),(170,205,177,198),(171,206,178,199),(172,207,179,200),(173,208,180,201),(174,209,181,202),(175,210,182,203),(183,218,190,211),(184,219,191,212),(185,220,192,213),(186,221,193,214),(187,222,194,215),(188,223,195,216),(189,224,196,217)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,29),(9,30),(10,31),(11,32),(12,33),(13,34),(14,35),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,56),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,141),(121,142),(122,143),(123,144),(124,145),(125,146),(126,147),(127,162),(128,163),(129,164),(130,165),(131,166),(132,167),(133,168),(134,155),(135,156),(136,157),(137,158),(138,159),(139,160),(140,161),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(183,190),(184,191),(185,192),(186,193),(187,194),(188,195),(189,196)], [(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)], [(1,194,8,187),(2,193,9,186),(3,192,10,185),(4,191,11,184),(5,190,12,183),(6,196,13,189),(7,195,14,188),(15,180,22,173),(16,179,23,172),(17,178,24,171),(18,177,25,170),(19,176,26,169),(20,182,27,175),(21,181,28,174),(29,222,36,215),(30,221,37,214),(31,220,38,213),(32,219,39,212),(33,218,40,211),(34,224,41,217),(35,223,42,216),(43,208,50,201),(44,207,51,200),(45,206,52,199),(46,205,53,198),(47,204,54,197),(48,210,55,203),(49,209,56,202),(57,124,64,117),(58,123,65,116),(59,122,66,115),(60,121,67,114),(61,120,68,113),(62,126,69,119),(63,125,70,118),(71,138,78,131),(72,137,79,130),(73,136,80,129),(74,135,81,128),(75,134,82,127),(76,140,83,133),(77,139,84,132),(85,152,92,145),(86,151,93,144),(87,150,94,143),(88,149,95,142),(89,148,96,141),(90,154,97,147),(91,153,98,146),(99,166,106,159),(100,165,107,158),(101,164,108,157),(102,163,109,156),(103,162,110,155),(104,168,111,161),(105,167,112,160)], [(1,71,15,57),(2,72,16,58),(3,73,17,59),(4,74,18,60),(5,75,19,61),(6,76,20,62),(7,77,21,63),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98),(113,190,127,176),(114,191,128,177),(115,192,129,178),(116,193,130,179),(117,194,131,180),(118,195,132,181),(119,196,133,182),(120,183,134,169),(121,184,135,170),(122,185,136,171),(123,186,137,172),(124,187,138,173),(125,188,139,174),(126,189,140,175),(141,218,155,204),(142,219,156,205),(143,220,157,206),(144,221,158,207),(145,222,159,208),(146,223,160,209),(147,224,161,210),(148,211,162,197),(149,212,163,198),(150,213,164,199),(151,214,165,200),(152,215,166,201),(153,216,167,202),(154,217,168,203)]])
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 14 | 14 | 14 | 14 | 28 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4×D7 | C8⋊C22 | C8.C22 | D4⋊2D7 | D4×D7 | D8⋊D7 | SD16⋊D7 |
| kernel | D4.D7⋊C4 | C28.Q8 | C56⋊C4 | C28.44D4 | C7×D4⋊C4 | Dic7⋊3Q8 | C2×D4.D7 | D4×Dic7 | D4.D7 | C2×Dic7 | D4⋊C4 | C28 | C4⋊C4 | C2×C8 | C2×D4 | D4 | C14 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 3 | 2 | 3 | 3 | 3 | 12 | 1 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of D4.D7⋊C4 ►in GL8(𝔽113)
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 49 | 112 |
| 0 | 0 | 0 | 0 | 9 | 112 | 29 | 104 |
| 0 | 0 | 0 | 0 | 40 | 96 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 71 | 49 | 0 |
| 112 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 56 | 111 | 1 | 0 | 0 | 0 | 0 | 0 |
| 37 | 1 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 49 | 112 |
| 0 | 0 | 0 | 0 | 9 | 112 | 29 | 104 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 | 49 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 93 | 83 | 10 | 112 | 0 | 0 | 0 | 0 |
| 63 | 59 | 11 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 64 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 9 | 112 |
| 0 | 0 | 0 | 0 | 58 | 112 | 1 | 0 |
| 66 | 52 | 34 | 59 | 0 | 0 | 0 | 0 |
| 9 | 22 | 5 | 54 | 0 | 0 | 0 | 0 |
| 111 | 64 | 70 | 97 | 0 | 0 | 0 | 0 |
| 19 | 79 | 103 | 68 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 76 | 106 | 68 | 13 |
| 0 | 0 | 0 | 0 | 14 | 24 | 2 | 4 |
| 0 | 0 | 0 | 0 | 68 | 5 | 26 | 0 |
| 0 | 0 | 0 | 0 | 68 | 19 | 76 | 100 |
| 40 | 83 | 9 | 71 | 0 | 0 | 0 | 0 |
| 31 | 7 | 33 | 80 | 0 | 0 | 0 | 0 |
| 65 | 46 | 7 | 13 | 0 | 0 | 0 | 0 |
| 3 | 57 | 43 | 59 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 87 | 84 | 66 | 52 |
| 0 | 0 | 0 | 0 | 39 | 87 | 18 | 31 |
| 0 | 0 | 0 | 0 | 46 | 20 | 39 | 29 |
| 0 | 0 | 0 | 0 | 46 | 76 | 37 | 13 |
G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,0,9,40,40,0,0,0,0,0,112,96,71,0,0,0,0,49,29,1,49,0,0,0,0,112,104,0,0],[112,0,56,37,0,0,0,0,0,112,111,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,112,0,0,0,0,0,112,0,0,0,0,0,0,49,29,1,49,0,0,0,0,112,104,0,0],[0,1,93,63,0,0,0,0,112,9,83,59,0,0,0,0,0,0,10,11,0,0,0,0,0,0,112,112,0,0,0,0,0,0,0,0,58,18,1,58,0,0,0,0,112,64,0,112,0,0,0,0,0,0,9,1,0,0,0,0,0,0,112,0],[66,9,111,19,0,0,0,0,52,22,64,79,0,0,0,0,34,5,70,103,0,0,0,0,59,54,97,68,0,0,0,0,0,0,0,0,76,14,68,68,0,0,0,0,106,24,5,19,0,0,0,0,68,2,26,76,0,0,0,0,13,4,0,100],[40,31,65,3,0,0,0,0,83,7,46,57,0,0,0,0,9,33,7,43,0,0,0,0,71,80,13,59,0,0,0,0,0,0,0,0,87,39,46,46,0,0,0,0,84,87,20,76,0,0,0,0,66,18,39,37,0,0,0,0,52,31,29,13] >;
D4.D7⋊C4 in GAP, Magma, Sage, TeX
D_4.D_7\rtimes C_4
% in TeX
G:=Group("D4.D7:C4"); // GroupNames label
G:=SmallGroup(448,291);
// by ID
G=gap.SmallGroup(448,291);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,758,135,100,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^7=e^4=1,d^2=a^2,b*a*b=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=e*b*e^-1=a*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations