direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D7×Q8⋊C4, D14.3Q16, D14.6SD16, Q8⋊5(C4×D7), (Q8×D7)⋊1C4, C2.2(D7×Q16), (C4×D7).39D4, C4.162(D4×D7), C2.4(D7×SD16), C4⋊C4.147D14, Dic14⋊4(C2×C4), (C2×C8).208D14, Q8⋊Dic7⋊3C2, C14.Q16⋊9C2, C28.116(C2×D4), C14.16(C2×Q16), C22.76(D4×D7), C28.12(C22×C4), (C2×Q8).103D14, (C2×Dic7).93D4, C14.29(C2×SD16), C28.44D4⋊22C2, (C2×C56).196C22, (C2×C28).241C23, (C22×D7).105D4, C4⋊Dic7.89C22, (Q8×C14).24C22, D14.19(C22⋊C4), Dic7.8(C22⋊C4), (C2×Dic14).68C22, C4.12(C2×C4×D7), (C2×Q8×D7).2C2, (D7×C4⋊C4).1C2, (C7×Q8)⋊2(C2×C4), C7⋊1(C2×Q8⋊C4), (D7×C2×C8).11C2, (C4×D7).14(C2×C4), C2.21(D7×C22⋊C4), (C7×Q8⋊C4)⋊18C2, (C2×C14).254(C2×D4), (C7×C4⋊C4).42C22, C14.20(C2×C22⋊C4), (C2×C7⋊C8).218C22, (C2×C4×D7).226C22, (C2×C4).348(C22×D7), SmallGroup(448,335)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D7×Q8⋊C4
G = < a,b,c,d,e | a7=b2=c4=e4=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece-1=c-1, ede-1=c-1d >
Subgroups: 788 in 162 conjugacy classes, 63 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, Q8, Q8, C23, D7, C14, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C2×Q8, Dic7, Dic7, C28, C28, D14, C2×C14, Q8⋊C4, Q8⋊C4, C2×C4⋊C4, C22×C8, C22×Q8, C7⋊C8, C56, Dic14, Dic14, C4×D7, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C2×Q8⋊C4, C8×D7, C2×C7⋊C8, Dic7⋊C4, C4⋊Dic7, C7×C4⋊C4, C2×C56, C2×Dic14, C2×Dic14, C2×C4×D7, C2×C4×D7, Q8×D7, Q8×D7, Q8×C14, C14.Q16, C28.44D4, Q8⋊Dic7, C7×Q8⋊C4, D7×C4⋊C4, D7×C2×C8, C2×Q8×D7, D7×Q8⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, SD16, Q16, C22×C4, C2×D4, D14, Q8⋊C4, C2×C22⋊C4, C2×SD16, C2×Q16, C4×D7, C22×D7, C2×Q8⋊C4, C2×C4×D7, D4×D7, D7×C22⋊C4, D7×SD16, D7×Q16, D7×Q8⋊C4
►On 224 points
Generators in S224
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)(113 115)(116 119)(117 118)(120 122)(123 126)(124 125)(127 129)(130 133)(131 132)(134 136)(137 140)(138 139)(141 143)(144 147)(145 146)(148 150)(151 154)(152 153)(155 157)(158 161)(159 160)(162 164)(165 168)(166 167)(169 171)(172 175)(173 174)(176 178)(179 182)(180 181)(183 185)(186 189)(187 188)(190 192)(193 196)(194 195)(197 199)(200 203)(201 202)(204 206)(207 210)(208 209)(211 213)(214 217)(215 216)(218 220)(221 224)(222 223)
(1 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)(113 134 120 127)(114 135 121 128)(115 136 122 129)(116 137 123 130)(117 138 124 131)(118 139 125 132)(119 140 126 133)(141 162 148 155)(142 163 149 156)(143 164 150 157)(144 165 151 158)(145 166 152 159)(146 167 153 160)(147 168 154 161)(169 183 176 190)(170 184 177 191)(171 185 178 192)(172 186 179 193)(173 187 180 194)(174 188 181 195)(175 189 182 196)(197 211 204 218)(198 212 205 219)(199 213 206 220)(200 214 207 221)(201 215 208 222)(202 216 209 223)(203 217 210 224)
(1 153 13 146)(2 154 14 147)(3 148 8 141)(4 149 9 142)(5 150 10 143)(6 151 11 144)(7 152 12 145)(15 162 22 155)(16 163 23 156)(17 164 24 157)(18 165 25 158)(19 166 26 159)(20 167 27 160)(21 168 28 161)(29 120 36 113)(30 121 37 114)(31 122 38 115)(32 123 39 116)(33 124 40 117)(34 125 41 118)(35 126 42 119)(43 134 50 127)(44 135 51 128)(45 136 52 129)(46 137 53 130)(47 138 54 131)(48 139 55 132)(49 140 56 133)(57 204 64 197)(58 205 65 198)(59 206 66 199)(60 207 67 200)(61 208 68 201)(62 209 69 202)(63 210 70 203)(71 218 78 211)(72 219 79 212)(73 220 80 213)(74 221 81 214)(75 222 82 215)(76 223 83 216)(77 224 84 217)(85 176 92 169)(86 177 93 170)(87 178 94 171)(88 179 95 172)(89 180 96 173)(90 181 97 174)(91 182 98 175)(99 190 106 183)(100 191 107 184)(101 192 108 185)(102 193 109 186)(103 194 110 187)(104 195 111 188)(105 196 112 189)
(1 90 34 62)(2 91 35 63)(3 85 29 57)(4 86 30 58)(5 87 31 59)(6 88 32 60)(7 89 33 61)(8 92 36 64)(9 93 37 65)(10 94 38 66)(11 95 39 67)(12 96 40 68)(13 97 41 69)(14 98 42 70)(15 99 43 71)(16 100 44 72)(17 101 45 73)(18 102 46 74)(19 103 47 75)(20 104 48 76)(21 105 49 77)(22 106 50 78)(23 107 51 79)(24 108 52 80)(25 109 53 81)(26 110 54 82)(27 111 55 83)(28 112 56 84)(113 211 141 183)(114 212 142 184)(115 213 143 185)(116 214 144 186)(117 215 145 187)(118 216 146 188)(119 217 147 189)(120 218 148 190)(121 219 149 191)(122 220 150 192)(123 221 151 193)(124 222 152 194)(125 223 153 195)(126 224 154 196)(127 204 155 176)(128 205 156 177)(129 206 157 178)(130 207 158 179)(131 208 159 180)(132 209 160 181)(133 210 161 182)(134 197 162 169)(135 198 163 170)(136 199 164 171)(137 200 165 172)(138 201 166 173)(139 202 167 174)(140 203 168 175)
G:=sub<Sym(224)| (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,87)(88,91)(89,90)(92,94)(95,98)(96,97)(99,101)(102,105)(103,104)(106,108)(109,112)(110,111)(113,115)(116,119)(117,118)(120,122)(123,126)(124,125)(127,129)(130,133)(131,132)(134,136)(137,140)(138,139)(141,143)(144,147)(145,146)(148,150)(151,154)(152,153)(155,157)(158,161)(159,160)(162,164)(165,168)(166,167)(169,171)(172,175)(173,174)(176,178)(179,182)(180,181)(183,185)(186,189)(187,188)(190,192)(193,196)(194,195)(197,199)(200,203)(201,202)(204,206)(207,210)(208,209)(211,213)(214,217)(215,216)(218,220)(221,224)(222,223), (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,134,120,127)(114,135,121,128)(115,136,122,129)(116,137,123,130)(117,138,124,131)(118,139,125,132)(119,140,126,133)(141,162,148,155)(142,163,149,156)(143,164,150,157)(144,165,151,158)(145,166,152,159)(146,167,153,160)(147,168,154,161)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,153,13,146)(2,154,14,147)(3,148,8,141)(4,149,9,142)(5,150,10,143)(6,151,11,144)(7,152,12,145)(15,162,22,155)(16,163,23,156)(17,164,24,157)(18,165,25,158)(19,166,26,159)(20,167,27,160)(21,168,28,161)(29,120,36,113)(30,121,37,114)(31,122,38,115)(32,123,39,116)(33,124,40,117)(34,125,41,118)(35,126,42,119)(43,134,50,127)(44,135,51,128)(45,136,52,129)(46,137,53,130)(47,138,54,131)(48,139,55,132)(49,140,56,133)(57,204,64,197)(58,205,65,198)(59,206,66,199)(60,207,67,200)(61,208,68,201)(62,209,69,202)(63,210,70,203)(71,218,78,211)(72,219,79,212)(73,220,80,213)(74,221,81,214)(75,222,82,215)(76,223,83,216)(77,224,84,217)(85,176,92,169)(86,177,93,170)(87,178,94,171)(88,179,95,172)(89,180,96,173)(90,181,97,174)(91,182,98,175)(99,190,106,183)(100,191,107,184)(101,192,108,185)(102,193,109,186)(103,194,110,187)(104,195,111,188)(105,196,112,189), (1,90,34,62)(2,91,35,63)(3,85,29,57)(4,86,30,58)(5,87,31,59)(6,88,32,60)(7,89,33,61)(8,92,36,64)(9,93,37,65)(10,94,38,66)(11,95,39,67)(12,96,40,68)(13,97,41,69)(14,98,42,70)(15,99,43,71)(16,100,44,72)(17,101,45,73)(18,102,46,74)(19,103,47,75)(20,104,48,76)(21,105,49,77)(22,106,50,78)(23,107,51,79)(24,108,52,80)(25,109,53,81)(26,110,54,82)(27,111,55,83)(28,112,56,84)(113,211,141,183)(114,212,142,184)(115,213,143,185)(116,214,144,186)(117,215,145,187)(118,216,146,188)(119,217,147,189)(120,218,148,190)(121,219,149,191)(122,220,150,192)(123,221,151,193)(124,222,152,194)(125,223,153,195)(126,224,154,196)(127,204,155,176)(128,205,156,177)(129,206,157,178)(130,207,158,179)(131,208,159,180)(132,209,160,181)(133,210,161,182)(134,197,162,169)(135,198,163,170)(136,199,164,171)(137,200,165,172)(138,201,166,173)(139,202,167,174)(140,203,168,175)>;
G:=Group( (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,31)(32,35)(33,34)(36,38)(39,42)(40,41)(43,45)(46,49)(47,48)(50,52)(53,56)(54,55)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,87)(88,91)(89,90)(92,94)(95,98)(96,97)(99,101)(102,105)(103,104)(106,108)(109,112)(110,111)(113,115)(116,119)(117,118)(120,122)(123,126)(124,125)(127,129)(130,133)(131,132)(134,136)(137,140)(138,139)(141,143)(144,147)(145,146)(148,150)(151,154)(152,153)(155,157)(158,161)(159,160)(162,164)(165,168)(166,167)(169,171)(172,175)(173,174)(176,178)(179,182)(180,181)(183,185)(186,189)(187,188)(190,192)(193,196)(194,195)(197,199)(200,203)(201,202)(204,206)(207,210)(208,209)(211,213)(214,217)(215,216)(218,220)(221,224)(222,223), (1,20,13,27)(2,21,14,28)(3,15,8,22)(4,16,9,23)(5,17,10,24)(6,18,11,25)(7,19,12,26)(29,43,36,50)(30,44,37,51)(31,45,38,52)(32,46,39,53)(33,47,40,54)(34,48,41,55)(35,49,42,56)(57,78,64,71)(58,79,65,72)(59,80,66,73)(60,81,67,74)(61,82,68,75)(62,83,69,76)(63,84,70,77)(85,106,92,99)(86,107,93,100)(87,108,94,101)(88,109,95,102)(89,110,96,103)(90,111,97,104)(91,112,98,105)(113,134,120,127)(114,135,121,128)(115,136,122,129)(116,137,123,130)(117,138,124,131)(118,139,125,132)(119,140,126,133)(141,162,148,155)(142,163,149,156)(143,164,150,157)(144,165,151,158)(145,166,152,159)(146,167,153,160)(147,168,154,161)(169,183,176,190)(170,184,177,191)(171,185,178,192)(172,186,179,193)(173,187,180,194)(174,188,181,195)(175,189,182,196)(197,211,204,218)(198,212,205,219)(199,213,206,220)(200,214,207,221)(201,215,208,222)(202,216,209,223)(203,217,210,224), (1,153,13,146)(2,154,14,147)(3,148,8,141)(4,149,9,142)(5,150,10,143)(6,151,11,144)(7,152,12,145)(15,162,22,155)(16,163,23,156)(17,164,24,157)(18,165,25,158)(19,166,26,159)(20,167,27,160)(21,168,28,161)(29,120,36,113)(30,121,37,114)(31,122,38,115)(32,123,39,116)(33,124,40,117)(34,125,41,118)(35,126,42,119)(43,134,50,127)(44,135,51,128)(45,136,52,129)(46,137,53,130)(47,138,54,131)(48,139,55,132)(49,140,56,133)(57,204,64,197)(58,205,65,198)(59,206,66,199)(60,207,67,200)(61,208,68,201)(62,209,69,202)(63,210,70,203)(71,218,78,211)(72,219,79,212)(73,220,80,213)(74,221,81,214)(75,222,82,215)(76,223,83,216)(77,224,84,217)(85,176,92,169)(86,177,93,170)(87,178,94,171)(88,179,95,172)(89,180,96,173)(90,181,97,174)(91,182,98,175)(99,190,106,183)(100,191,107,184)(101,192,108,185)(102,193,109,186)(103,194,110,187)(104,195,111,188)(105,196,112,189), (1,90,34,62)(2,91,35,63)(3,85,29,57)(4,86,30,58)(5,87,31,59)(6,88,32,60)(7,89,33,61)(8,92,36,64)(9,93,37,65)(10,94,38,66)(11,95,39,67)(12,96,40,68)(13,97,41,69)(14,98,42,70)(15,99,43,71)(16,100,44,72)(17,101,45,73)(18,102,46,74)(19,103,47,75)(20,104,48,76)(21,105,49,77)(22,106,50,78)(23,107,51,79)(24,108,52,80)(25,109,53,81)(26,110,54,82)(27,111,55,83)(28,112,56,84)(113,211,141,183)(114,212,142,184)(115,213,143,185)(116,214,144,186)(117,215,145,187)(118,216,146,188)(119,217,147,189)(120,218,148,190)(121,219,149,191)(122,220,150,192)(123,221,151,193)(124,222,152,194)(125,223,153,195)(126,224,154,196)(127,204,155,176)(128,205,156,177)(129,206,157,178)(130,207,158,179)(131,208,159,180)(132,209,160,181)(133,210,161,182)(134,197,162,169)(135,198,163,170)(136,199,164,171)(137,200,165,172)(138,201,166,173)(139,202,167,174)(140,203,168,175) );
G=PermutationGroup([[(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,31),(32,35),(33,34),(36,38),(39,42),(40,41),(43,45),(46,49),(47,48),(50,52),(53,56),(54,55),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83),(85,87),(88,91),(89,90),(92,94),(95,98),(96,97),(99,101),(102,105),(103,104),(106,108),(109,112),(110,111),(113,115),(116,119),(117,118),(120,122),(123,126),(124,125),(127,129),(130,133),(131,132),(134,136),(137,140),(138,139),(141,143),(144,147),(145,146),(148,150),(151,154),(152,153),(155,157),(158,161),(159,160),(162,164),(165,168),(166,167),(169,171),(172,175),(173,174),(176,178),(179,182),(180,181),(183,185),(186,189),(187,188),(190,192),(193,196),(194,195),(197,199),(200,203),(201,202),(204,206),(207,210),(208,209),(211,213),(214,217),(215,216),(218,220),(221,224),(222,223)], [(1,20,13,27),(2,21,14,28),(3,15,8,22),(4,16,9,23),(5,17,10,24),(6,18,11,25),(7,19,12,26),(29,43,36,50),(30,44,37,51),(31,45,38,52),(32,46,39,53),(33,47,40,54),(34,48,41,55),(35,49,42,56),(57,78,64,71),(58,79,65,72),(59,80,66,73),(60,81,67,74),(61,82,68,75),(62,83,69,76),(63,84,70,77),(85,106,92,99),(86,107,93,100),(87,108,94,101),(88,109,95,102),(89,110,96,103),(90,111,97,104),(91,112,98,105),(113,134,120,127),(114,135,121,128),(115,136,122,129),(116,137,123,130),(117,138,124,131),(118,139,125,132),(119,140,126,133),(141,162,148,155),(142,163,149,156),(143,164,150,157),(144,165,151,158),(145,166,152,159),(146,167,153,160),(147,168,154,161),(169,183,176,190),(170,184,177,191),(171,185,178,192),(172,186,179,193),(173,187,180,194),(174,188,181,195),(175,189,182,196),(197,211,204,218),(198,212,205,219),(199,213,206,220),(200,214,207,221),(201,215,208,222),(202,216,209,223),(203,217,210,224)], [(1,153,13,146),(2,154,14,147),(3,148,8,141),(4,149,9,142),(5,150,10,143),(6,151,11,144),(7,152,12,145),(15,162,22,155),(16,163,23,156),(17,164,24,157),(18,165,25,158),(19,166,26,159),(20,167,27,160),(21,168,28,161),(29,120,36,113),(30,121,37,114),(31,122,38,115),(32,123,39,116),(33,124,40,117),(34,125,41,118),(35,126,42,119),(43,134,50,127),(44,135,51,128),(45,136,52,129),(46,137,53,130),(47,138,54,131),(48,139,55,132),(49,140,56,133),(57,204,64,197),(58,205,65,198),(59,206,66,199),(60,207,67,200),(61,208,68,201),(62,209,69,202),(63,210,70,203),(71,218,78,211),(72,219,79,212),(73,220,80,213),(74,221,81,214),(75,222,82,215),(76,223,83,216),(77,224,84,217),(85,176,92,169),(86,177,93,170),(87,178,94,171),(88,179,95,172),(89,180,96,173),(90,181,97,174),(91,182,98,175),(99,190,106,183),(100,191,107,184),(101,192,108,185),(102,193,109,186),(103,194,110,187),(104,195,111,188),(105,196,112,189)], [(1,90,34,62),(2,91,35,63),(3,85,29,57),(4,86,30,58),(5,87,31,59),(6,88,32,60),(7,89,33,61),(8,92,36,64),(9,93,37,65),(10,94,38,66),(11,95,39,67),(12,96,40,68),(13,97,41,69),(14,98,42,70),(15,99,43,71),(16,100,44,72),(17,101,45,73),(18,102,46,74),(19,103,47,75),(20,104,48,76),(21,105,49,77),(22,106,50,78),(23,107,51,79),(24,108,52,80),(25,109,53,81),(26,110,54,82),(27,111,55,83),(28,112,56,84),(113,211,141,183),(114,212,142,184),(115,213,143,185),(116,214,144,186),(117,215,145,187),(118,216,146,188),(119,217,147,189),(120,218,148,190),(121,219,149,191),(122,220,150,192),(123,221,151,193),(124,222,152,194),(125,223,153,195),(126,224,154,196),(127,204,155,176),(128,205,156,177),(129,206,157,178),(130,207,158,179),(131,208,159,180),(132,209,160,181),(133,210,161,182),(134,197,162,169),(135,198,163,170),(136,199,164,171),(137,200,165,172),(138,201,166,173),(139,202,167,174),(140,203,168,175)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D4 | D7 | SD16 | Q16 | D14 | D14 | D14 | C4×D7 | D4×D7 | D4×D7 | D7×SD16 | D7×Q16 |
| kernel | D7×Q8⋊C4 | C14.Q16 | C28.44D4 | Q8⋊Dic7 | C7×Q8⋊C4 | D7×C4⋊C4 | D7×C2×C8 | C2×Q8×D7 | Q8×D7 | C4×D7 | C2×Dic7 | C22×D7 | Q8⋊C4 | D14 | D14 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 2 | 1 | 1 | 3 | 4 | 4 | 3 | 3 | 3 | 12 | 3 | 3 | 6 | 6 |
Matrix representation of D7×Q8⋊C4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 79 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 26 | 0 | 0 |
| 26 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 26 | 0 | 0 |
| 111 | 86 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 51 | 36 | 0 | 0 |
| 47 | 62 | 0 | 0 |
| 0 | 0 | 98 | 0 |
| 0 | 0 | 0 | 98 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,79],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[1,26,0,0,26,112,0,0,0,0,1,0,0,0,0,1],[27,111,0,0,26,86,0,0,0,0,1,0,0,0,0,1],[51,47,0,0,36,62,0,0,0,0,98,0,0,0,0,98] >;
D7×Q8⋊C4 in GAP, Magma, Sage, TeX
D_7\times Q_8\rtimes C_4
% in TeX
G:=Group("D7xQ8:C4"); // GroupNames label
G:=SmallGroup(448,335);
// by ID
G=gap.SmallGroup(448,335);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,219,58,851,438,102,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^4=e^4=1,d^2=c^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,d*c*d^-1=e*c*e^-1=c^-1,e*d*e^-1=c^-1*d>;
// generators/relations