metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C7×D4).8D4, (D4×Dic7)⋊7C2, (C2×SD16)⋊8D7, Dic7⋊C8⋊34C2, C28.172(C2×D4), (C2×C8).145D14, C7⋊7(D4.2D4), D4.3(C7⋊D4), (C2×Q8).51D14, C2.D56⋊34C2, (C14×SD16)⋊19C2, (C2×D4).145D14, C14.60(C4○D8), C28.99(C4○D4), C28.23D4⋊3C2, Q8⋊Dic7⋊27C2, (C2×Dic7).68D4, C22.262(D4×D7), C4.11(D4⋊2D7), C2.27(D56⋊C2), C14.77(C8⋊C22), (C2×C56).292C22, (C2×C28).442C23, (D4×C14).91C22, (Q8×C14).72C22, C14.114(C4⋊D4), (C2×D28).118C22, C4⋊Dic7.172C22, (C4×Dic7).49C22, C2.26(SD16⋊3D7), C2.26(Dic7⋊D4), (C2×D4⋊D7).8C2, C4.40(C2×C7⋊D4), (C2×C14).354(C2×D4), (C2×C7⋊C8).154C22, (C2×C4).531(C22×D7), SmallGroup(448,699)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C7×D4).D4
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, dad-1=eae=a-1, cbc=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=b2d-1 >
Subgroups: 676 in 124 conjugacy classes, 41 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D7, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, Dic7, C28, C28, D14, C2×C14, C2×C14, D4⋊C4, Q8⋊C4, C4⋊C8, C4×D4, C4.4D4, C2×D8, C2×SD16, C7⋊C8, C56, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×D7, C22×C14, D4.2D4, C2×C7⋊C8, C4×Dic7, C4⋊Dic7, D14⋊C4, D4⋊D7, C23.D7, C2×C56, C7×SD16, C2×D28, C22×Dic7, D4×C14, Q8×C14, Dic7⋊C8, C2.D56, Q8⋊Dic7, C2×D4⋊D7, D4×Dic7, C28.23D4, C14×SD16, (C7×D4).D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C4○D8, C8⋊C22, C7⋊D4, C22×D7, D4.2D4, D4×D7, D4⋊2D7, C2×C7⋊D4, D56⋊C2, SD16⋊3D7, Dic7⋊D4, (C7×D4).D4
►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 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 85 64 92)(58 86 65 93)(59 87 66 94)(60 88 67 95)(61 89 68 96)(62 90 69 97)(63 91 70 98)(71 99 78 106)(72 100 79 107)(73 101 80 108)(74 102 81 109)(75 103 82 110)(76 104 83 111)(77 105 84 112)(113 148 120 141)(114 149 121 142)(115 150 122 143)(116 151 123 144)(117 152 124 145)(118 153 125 146)(119 154 126 147)(127 162 134 155)(128 163 135 156)(129 164 136 157)(130 165 137 158)(131 166 138 159)(132 167 139 160)(133 168 140 161)(169 197 176 204)(170 198 177 205)(171 199 178 206)(172 200 179 207)(173 201 180 208)(174 202 181 209)(175 203 182 210)(183 211 190 218)(184 212 191 219)(185 213 192 220)(186 214 193 221)(187 215 194 222)(188 216 195 223)(189 217 196 224)
(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)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)(113 141)(114 142)(115 143)(116 144)(117 145)(118 146)(119 147)(120 148)(121 149)(122 150)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 159)(132 160)(133 161)(134 162)(135 163)(136 164)(137 165)(138 166)(139 167)(140 168)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)(211 218)(212 219)(213 220)(214 221)(215 222)(216 223)(217 224)
(1 132 15 118)(2 131 16 117)(3 130 17 116)(4 129 18 115)(5 128 19 114)(6 127 20 113)(7 133 21 119)(8 139 22 125)(9 138 23 124)(10 137 24 123)(11 136 25 122)(12 135 26 121)(13 134 27 120)(14 140 28 126)(29 160 43 146)(30 159 44 145)(31 158 45 144)(32 157 46 143)(33 156 47 142)(34 155 48 141)(35 161 49 147)(36 167 50 153)(37 166 51 152)(38 165 52 151)(39 164 53 150)(40 163 54 149)(41 162 55 148)(42 168 56 154)(57 188 71 174)(58 187 72 173)(59 186 73 172)(60 185 74 171)(61 184 75 170)(62 183 76 169)(63 189 77 175)(64 195 78 181)(65 194 79 180)(66 193 80 179)(67 192 81 178)(68 191 82 177)(69 190 83 176)(70 196 84 182)(85 216 99 202)(86 215 100 201)(87 214 101 200)(88 213 102 199)(89 212 103 198)(90 211 104 197)(91 217 105 203)(92 223 106 209)(93 222 107 208)(94 221 108 207)(95 220 109 206)(96 219 110 205)(97 218 111 204)(98 224 112 210)
(1 174)(2 173)(3 172)(4 171)(5 170)(6 169)(7 175)(8 181)(9 180)(10 179)(11 178)(12 177)(13 176)(14 182)(15 188)(16 187)(17 186)(18 185)(19 184)(20 183)(21 189)(22 195)(23 194)(24 193)(25 192)(26 191)(27 190)(28 196)(29 202)(30 201)(31 200)(32 199)(33 198)(34 197)(35 203)(36 209)(37 208)(38 207)(39 206)(40 205)(41 204)(42 210)(43 216)(44 215)(45 214)(46 213)(47 212)(48 211)(49 217)(50 223)(51 222)(52 221)(53 220)(54 219)(55 218)(56 224)(57 125)(58 124)(59 123)(60 122)(61 121)(62 120)(63 126)(64 118)(65 117)(66 116)(67 115)(68 114)(69 113)(70 119)(71 139)(72 138)(73 137)(74 136)(75 135)(76 134)(77 140)(78 132)(79 131)(80 130)(81 129)(82 128)(83 127)(84 133)(85 153)(86 152)(87 151)(88 150)(89 149)(90 148)(91 154)(92 146)(93 145)(94 144)(95 143)(96 142)(97 141)(98 147)(99 167)(100 166)(101 165)(102 164)(103 163)(104 162)(105 168)(106 160)(107 159)(108 158)(109 157)(110 156)(111 155)(112 161)
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), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(71,99,78,106)(72,100,79,107)(73,101,80,108)(74,102,81,109)(75,103,82,110)(76,104,83,111)(77,105,84,112)(113,148,120,141)(114,149,121,142)(115,150,122,143)(116,151,123,144)(117,152,124,145)(118,153,125,146)(119,154,126,147)(127,162,134,155)(128,163,135,156)(129,164,136,157)(130,165,137,158)(131,166,138,159)(132,167,139,160)(133,168,140,161)(169,197,176,204)(170,198,177,205)(171,199,178,206)(172,200,179,207)(173,201,180,208)(174,202,181,209)(175,203,182,210)(183,211,190,218)(184,212,191,219)(185,213,192,220)(186,214,193,221)(187,215,194,222)(188,216,195,223)(189,217,196,224), (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)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,132,15,118)(2,131,16,117)(3,130,17,116)(4,129,18,115)(5,128,19,114)(6,127,20,113)(7,133,21,119)(8,139,22,125)(9,138,23,124)(10,137,24,123)(11,136,25,122)(12,135,26,121)(13,134,27,120)(14,140,28,126)(29,160,43,146)(30,159,44,145)(31,158,45,144)(32,157,46,143)(33,156,47,142)(34,155,48,141)(35,161,49,147)(36,167,50,153)(37,166,51,152)(38,165,52,151)(39,164,53,150)(40,163,54,149)(41,162,55,148)(42,168,56,154)(57,188,71,174)(58,187,72,173)(59,186,73,172)(60,185,74,171)(61,184,75,170)(62,183,76,169)(63,189,77,175)(64,195,78,181)(65,194,79,180)(66,193,80,179)(67,192,81,178)(68,191,82,177)(69,190,83,176)(70,196,84,182)(85,216,99,202)(86,215,100,201)(87,214,101,200)(88,213,102,199)(89,212,103,198)(90,211,104,197)(91,217,105,203)(92,223,106,209)(93,222,107,208)(94,221,108,207)(95,220,109,206)(96,219,110,205)(97,218,111,204)(98,224,112,210), (1,174)(2,173)(3,172)(4,171)(5,170)(6,169)(7,175)(8,181)(9,180)(10,179)(11,178)(12,177)(13,176)(14,182)(15,188)(16,187)(17,186)(18,185)(19,184)(20,183)(21,189)(22,195)(23,194)(24,193)(25,192)(26,191)(27,190)(28,196)(29,202)(30,201)(31,200)(32,199)(33,198)(34,197)(35,203)(36,209)(37,208)(38,207)(39,206)(40,205)(41,204)(42,210)(43,216)(44,215)(45,214)(46,213)(47,212)(48,211)(49,217)(50,223)(51,222)(52,221)(53,220)(54,219)(55,218)(56,224)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,126)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,119)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,140)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,133)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,154)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,147)(99,167)(100,166)(101,165)(102,164)(103,163)(104,162)(105,168)(106,160)(107,159)(108,158)(109,157)(110,156)(111,155)(112,161)>;
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), (1,36,8,29)(2,37,9,30)(3,38,10,31)(4,39,11,32)(5,40,12,33)(6,41,13,34)(7,42,14,35)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,85,64,92)(58,86,65,93)(59,87,66,94)(60,88,67,95)(61,89,68,96)(62,90,69,97)(63,91,70,98)(71,99,78,106)(72,100,79,107)(73,101,80,108)(74,102,81,109)(75,103,82,110)(76,104,83,111)(77,105,84,112)(113,148,120,141)(114,149,121,142)(115,150,122,143)(116,151,123,144)(117,152,124,145)(118,153,125,146)(119,154,126,147)(127,162,134,155)(128,163,135,156)(129,164,136,157)(130,165,137,158)(131,166,138,159)(132,167,139,160)(133,168,140,161)(169,197,176,204)(170,198,177,205)(171,199,178,206)(172,200,179,207)(173,201,180,208)(174,202,181,209)(175,203,182,210)(183,211,190,218)(184,212,191,219)(185,213,192,220)(186,214,193,221)(187,215,194,222)(188,216,195,223)(189,217,196,224), (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)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(99,106)(100,107)(101,108)(102,109)(103,110)(104,111)(105,112)(113,141)(114,142)(115,143)(116,144)(117,145)(118,146)(119,147)(120,148)(121,149)(122,150)(123,151)(124,152)(125,153)(126,154)(127,155)(128,156)(129,157)(130,158)(131,159)(132,160)(133,161)(134,162)(135,163)(136,164)(137,165)(138,166)(139,167)(140,168)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210)(211,218)(212,219)(213,220)(214,221)(215,222)(216,223)(217,224), (1,132,15,118)(2,131,16,117)(3,130,17,116)(4,129,18,115)(5,128,19,114)(6,127,20,113)(7,133,21,119)(8,139,22,125)(9,138,23,124)(10,137,24,123)(11,136,25,122)(12,135,26,121)(13,134,27,120)(14,140,28,126)(29,160,43,146)(30,159,44,145)(31,158,45,144)(32,157,46,143)(33,156,47,142)(34,155,48,141)(35,161,49,147)(36,167,50,153)(37,166,51,152)(38,165,52,151)(39,164,53,150)(40,163,54,149)(41,162,55,148)(42,168,56,154)(57,188,71,174)(58,187,72,173)(59,186,73,172)(60,185,74,171)(61,184,75,170)(62,183,76,169)(63,189,77,175)(64,195,78,181)(65,194,79,180)(66,193,80,179)(67,192,81,178)(68,191,82,177)(69,190,83,176)(70,196,84,182)(85,216,99,202)(86,215,100,201)(87,214,101,200)(88,213,102,199)(89,212,103,198)(90,211,104,197)(91,217,105,203)(92,223,106,209)(93,222,107,208)(94,221,108,207)(95,220,109,206)(96,219,110,205)(97,218,111,204)(98,224,112,210), (1,174)(2,173)(3,172)(4,171)(5,170)(6,169)(7,175)(8,181)(9,180)(10,179)(11,178)(12,177)(13,176)(14,182)(15,188)(16,187)(17,186)(18,185)(19,184)(20,183)(21,189)(22,195)(23,194)(24,193)(25,192)(26,191)(27,190)(28,196)(29,202)(30,201)(31,200)(32,199)(33,198)(34,197)(35,203)(36,209)(37,208)(38,207)(39,206)(40,205)(41,204)(42,210)(43,216)(44,215)(45,214)(46,213)(47,212)(48,211)(49,217)(50,223)(51,222)(52,221)(53,220)(54,219)(55,218)(56,224)(57,125)(58,124)(59,123)(60,122)(61,121)(62,120)(63,126)(64,118)(65,117)(66,116)(67,115)(68,114)(69,113)(70,119)(71,139)(72,138)(73,137)(74,136)(75,135)(76,134)(77,140)(78,132)(79,131)(80,130)(81,129)(82,128)(83,127)(84,133)(85,153)(86,152)(87,151)(88,150)(89,149)(90,148)(91,154)(92,146)(93,145)(94,144)(95,143)(96,142)(97,141)(98,147)(99,167)(100,166)(101,165)(102,164)(103,163)(104,162)(105,168)(106,160)(107,159)(108,158)(109,157)(110,156)(111,155)(112,161) );
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)], [(1,36,8,29),(2,37,9,30),(3,38,10,31),(4,39,11,32),(5,40,12,33),(6,41,13,34),(7,42,14,35),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49),(57,85,64,92),(58,86,65,93),(59,87,66,94),(60,88,67,95),(61,89,68,96),(62,90,69,97),(63,91,70,98),(71,99,78,106),(72,100,79,107),(73,101,80,108),(74,102,81,109),(75,103,82,110),(76,104,83,111),(77,105,84,112),(113,148,120,141),(114,149,121,142),(115,150,122,143),(116,151,123,144),(117,152,124,145),(118,153,125,146),(119,154,126,147),(127,162,134,155),(128,163,135,156),(129,164,136,157),(130,165,137,158),(131,166,138,159),(132,167,139,160),(133,168,140,161),(169,197,176,204),(170,198,177,205),(171,199,178,206),(172,200,179,207),(173,201,180,208),(174,202,181,209),(175,203,182,210),(183,211,190,218),(184,212,191,219),(185,213,192,220),(186,214,193,221),(187,215,194,222),(188,216,195,223),(189,217,196,224)], [(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),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(99,106),(100,107),(101,108),(102,109),(103,110),(104,111),(105,112),(113,141),(114,142),(115,143),(116,144),(117,145),(118,146),(119,147),(120,148),(121,149),(122,150),(123,151),(124,152),(125,153),(126,154),(127,155),(128,156),(129,157),(130,158),(131,159),(132,160),(133,161),(134,162),(135,163),(136,164),(137,165),(138,166),(139,167),(140,168),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210),(211,218),(212,219),(213,220),(214,221),(215,222),(216,223),(217,224)], [(1,132,15,118),(2,131,16,117),(3,130,17,116),(4,129,18,115),(5,128,19,114),(6,127,20,113),(7,133,21,119),(8,139,22,125),(9,138,23,124),(10,137,24,123),(11,136,25,122),(12,135,26,121),(13,134,27,120),(14,140,28,126),(29,160,43,146),(30,159,44,145),(31,158,45,144),(32,157,46,143),(33,156,47,142),(34,155,48,141),(35,161,49,147),(36,167,50,153),(37,166,51,152),(38,165,52,151),(39,164,53,150),(40,163,54,149),(41,162,55,148),(42,168,56,154),(57,188,71,174),(58,187,72,173),(59,186,73,172),(60,185,74,171),(61,184,75,170),(62,183,76,169),(63,189,77,175),(64,195,78,181),(65,194,79,180),(66,193,80,179),(67,192,81,178),(68,191,82,177),(69,190,83,176),(70,196,84,182),(85,216,99,202),(86,215,100,201),(87,214,101,200),(88,213,102,199),(89,212,103,198),(90,211,104,197),(91,217,105,203),(92,223,106,209),(93,222,107,208),(94,221,108,207),(95,220,109,206),(96,219,110,205),(97,218,111,204),(98,224,112,210)], [(1,174),(2,173),(3,172),(4,171),(5,170),(6,169),(7,175),(8,181),(9,180),(10,179),(11,178),(12,177),(13,176),(14,182),(15,188),(16,187),(17,186),(18,185),(19,184),(20,183),(21,189),(22,195),(23,194),(24,193),(25,192),(26,191),(27,190),(28,196),(29,202),(30,201),(31,200),(32,199),(33,198),(34,197),(35,203),(36,209),(37,208),(38,207),(39,206),(40,205),(41,204),(42,210),(43,216),(44,215),(45,214),(46,213),(47,212),(48,211),(49,217),(50,223),(51,222),(52,221),(53,220),(54,219),(55,218),(56,224),(57,125),(58,124),(59,123),(60,122),(61,121),(62,120),(63,126),(64,118),(65,117),(66,116),(67,115),(68,114),(69,113),(70,119),(71,139),(72,138),(73,137),(74,136),(75,135),(76,134),(77,140),(78,132),(79,131),(80,130),(81,129),(82,128),(83,127),(84,133),(85,153),(86,152),(87,151),(88,150),(89,149),(90,148),(91,154),(92,146),(93,145),(94,144),(95,143),(96,142),(97,141),(98,147),(99,167),(100,166),(101,165),(102,164),(103,163),(104,162),(105,168),(106,160),(107,159),(108,158),(109,157),(110,156),(111,155),(112,161)]])
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 14J | ··· | 14O | 28A | ··· | 28F | 28G | ··· | 28L | 56A | ··· | 56L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 56 | 2 | 2 | 8 | 14 | 14 | 28 | 28 | 28 | 2 | 2 | 2 | 4 | 4 | 28 | 28 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D7 | C4○D4 | D14 | D14 | D14 | C4○D8 | C7⋊D4 | C8⋊C22 | D4⋊2D7 | D4×D7 | D56⋊C2 | SD16⋊3D7 |
| kernel | (C7×D4).D4 | Dic7⋊C8 | C2.D56 | Q8⋊Dic7 | C2×D4⋊D7 | D4×Dic7 | C28.23D4 | C14×SD16 | C2×Dic7 | C7×D4 | C2×SD16 | C28 | C2×C8 | C2×D4 | C2×Q8 | C14 | D4 | C14 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 2 | 3 | 3 | 3 | 4 | 12 | 1 | 3 | 3 | 6 | 6 |
Matrix representation of (C7×D4).D4 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 24 | 112 |
| 0 | 0 | 1 | 0 |
| 1 | 2 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 112 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 98 | 0 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 10 | 86 |
| 0 | 0 | 100 | 103 |
| 0 | 51 | 0 | 0 |
| 82 | 0 | 0 | 0 |
| 0 | 0 | 27 | 62 |
| 0 | 0 | 32 | 86 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,24,1,0,0,112,0],[1,112,0,0,2,112,0,0,0,0,1,0,0,0,0,1],[1,112,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[98,0,0,0,0,98,0,0,0,0,10,100,0,0,86,103],[0,82,0,0,51,0,0,0,0,0,27,32,0,0,62,86] >;
(C7×D4).D4 in GAP, Magma, Sage, TeX
(C_7\times D_4).D_4
% in TeX
G:=Group("(C7xD4).D4"); // GroupNames label
G:=SmallGroup(448,699);
// by ID
G=gap.SmallGroup(448,699);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,1094,135,184,570,297,136,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=b^2*d^-1>;
// generators/relations