direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×D4⋊6D4, C14.1172- 1+4, D4⋊6(C7×D4), (C7×D4)⋊24D4, C4⋊Q8⋊13C14, (C4×D4)⋊16C14, (D4×C28)⋊45C2, C4.41(D4×C14), C28⋊16(C4○D4), C4⋊D4⋊12C14, C28.402(C2×D4), C22⋊Q8⋊12C14, C22.6(D4×C14), C42.42(C2×C14), (C2×C28).675C23, (C2×C14).367C24, (C4×C28).283C22, C22.D4⋊9C14, C14.195(C22×D4), C2.9(C7×2- 1+4), (D4×C14).321C22, C23.41(C22×C14), C22.41(C23×C14), (C22×C14).99C23, (Q8×C14).273C22, (C22×C28).453C22, C4⋊2(C7×C4○D4), (C7×C4⋊Q8)⋊34C2, (C14×C4⋊C4)⋊47C2, (C2×C4⋊C4)⋊20C14, C2.19(D4×C2×C14), (C2×C4○D4)⋊7C14, (C14×C4○D4)⋊23C2, C4⋊C4.31(C2×C14), (C7×C4⋊D4)⋊39C2, C2.21(C14×C4○D4), (C7×C22⋊Q8)⋊39C2, (C2×D4).66(C2×C14), C14.240(C2×C4○D4), (C2×C14).183(C2×D4), C22⋊C4.5(C2×C14), (C2×Q8).60(C2×C14), (C7×C4⋊C4).395C22, (C22×C4).65(C2×C14), (C2×C4).33(C22×C14), (C7×C22.D4)⋊28C2, (C7×C22⋊C4).87C22, SmallGroup(448,1330)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×D4⋊6D4
G = < a,b,c,d,e | a7=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 426 in 292 conjugacy classes, 166 normal (34 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C28, C28, C2×C14, C2×C14, C2×C14, C2×C4⋊C4, C4×D4, C4⋊D4, C22⋊Q8, C22.D4, C4⋊Q8, C2×C4○D4, C2×C28, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C22×C14, D4⋊6D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C4⋊C4, C22×C28, D4×C14, D4×C14, Q8×C14, C7×C4○D4, C14×C4⋊C4, D4×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C22.D4, C7×C4⋊Q8, C14×C4○D4, C7×D4⋊6D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C24, C2×C14, C22×D4, C2×C4○D4, 2- 1+4, C7×D4, C22×C14, D4⋊6D4, D4×C14, C7×C4○D4, C23×C14, D4×C2×C14, C14×C4○D4, C7×2- 1+4, C7×D4⋊6D4
►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 63 34 67)(2 57 35 68)(3 58 29 69)(4 59 30 70)(5 60 31 64)(6 61 32 65)(7 62 33 66)(8 200 222 204)(9 201 223 205)(10 202 224 206)(11 203 218 207)(12 197 219 208)(13 198 220 209)(14 199 221 210)(15 212 22 195)(16 213 23 196)(17 214 24 190)(18 215 25 191)(19 216 26 192)(20 217 27 193)(21 211 28 194)(36 54 43 71)(37 55 44 72)(38 56 45 73)(39 50 46 74)(40 51 47 75)(41 52 48 76)(42 53 49 77)(78 123 102 119)(79 124 103 113)(80 125 104 114)(81 126 105 115)(82 120 99 116)(83 121 100 117)(84 122 101 118)(85 131 96 107)(86 132 97 108)(87 133 98 109)(88 127 92 110)(89 128 93 111)(90 129 94 112)(91 130 95 106)(134 175 158 179)(135 169 159 180)(136 170 160 181)(137 171 161 182)(138 172 155 176)(139 173 156 177)(140 174 157 178)(141 163 152 187)(142 164 153 188)(143 165 154 189)(144 166 148 183)(145 167 149 184)(146 168 150 185)(147 162 151 186)
(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 124)(114 125)(115 126)(116 120)(117 121)(118 122)(119 123)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 208)(198 209)(199 210)(200 204)(201 205)(202 206)(203 207)
(1 11 46 17)(2 12 47 18)(3 13 48 19)(4 14 49 20)(5 8 43 21)(6 9 44 15)(7 10 45 16)(22 32 223 37)(23 33 224 38)(24 34 218 39)(25 35 219 40)(26 29 220 41)(27 30 221 42)(28 31 222 36)(50 190 67 207)(51 191 68 208)(52 192 69 209)(53 193 70 210)(54 194 64 204)(55 195 65 205)(56 196 66 206)(57 197 75 215)(58 198 76 216)(59 199 77 217)(60 200 71 211)(61 201 72 212)(62 202 73 213)(63 203 74 214)(78 186 95 175)(79 187 96 169)(80 188 97 170)(81 189 98 171)(82 183 92 172)(83 184 93 173)(84 185 94 174)(85 180 103 163)(86 181 104 164)(87 182 105 165)(88 176 99 166)(89 177 100 167)(90 178 101 168)(91 179 102 162)(106 158 123 147)(107 159 124 141)(108 160 125 142)(109 161 126 143)(110 155 120 144)(111 156 121 145)(112 157 122 146)(113 152 131 135)(114 153 132 136)(115 154 133 137)(116 148 127 138)(117 149 128 139)(118 150 129 140)(119 151 130 134)
(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 110)(9 111)(10 112)(11 106)(12 107)(13 108)(14 109)(15 121)(16 122)(17 123)(18 124)(19 125)(20 126)(21 120)(22 117)(23 118)(24 119)(25 113)(26 114)(27 115)(28 116)(29 136)(30 137)(31 138)(32 139)(33 140)(34 134)(35 135)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 144)(44 145)(45 146)(46 147)(47 141)(48 142)(49 143)(50 186)(51 187)(52 188)(53 189)(54 183)(55 184)(56 185)(57 180)(58 181)(59 182)(60 176)(61 177)(62 178)(63 179)(64 172)(65 173)(66 174)(67 175)(68 169)(69 170)(70 171)(71 166)(72 167)(73 168)(74 162)(75 163)(76 164)(77 165)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(127 222)(128 223)(129 224)(130 218)(131 219)(132 220)(133 221)
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), (1,63,34,67)(2,57,35,68)(3,58,29,69)(4,59,30,70)(5,60,31,64)(6,61,32,65)(7,62,33,66)(8,200,222,204)(9,201,223,205)(10,202,224,206)(11,203,218,207)(12,197,219,208)(13,198,220,209)(14,199,221,210)(15,212,22,195)(16,213,23,196)(17,214,24,190)(18,215,25,191)(19,216,26,192)(20,217,27,193)(21,211,28,194)(36,54,43,71)(37,55,44,72)(38,56,45,73)(39,50,46,74)(40,51,47,75)(41,52,48,76)(42,53,49,77)(78,123,102,119)(79,124,103,113)(80,125,104,114)(81,126,105,115)(82,120,99,116)(83,121,100,117)(84,122,101,118)(85,131,96,107)(86,132,97,108)(87,133,98,109)(88,127,92,110)(89,128,93,111)(90,129,94,112)(91,130,95,106)(134,175,158,179)(135,169,159,180)(136,170,160,181)(137,171,161,182)(138,172,155,176)(139,173,156,177)(140,174,157,178)(141,163,152,187)(142,164,153,188)(143,165,154,189)(144,166,148,183)(145,167,149,184)(146,168,150,185)(147,162,151,186), (50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,208)(198,209)(199,210)(200,204)(201,205)(202,206)(203,207), (1,11,46,17)(2,12,47,18)(3,13,48,19)(4,14,49,20)(5,8,43,21)(6,9,44,15)(7,10,45,16)(22,32,223,37)(23,33,224,38)(24,34,218,39)(25,35,219,40)(26,29,220,41)(27,30,221,42)(28,31,222,36)(50,190,67,207)(51,191,68,208)(52,192,69,209)(53,193,70,210)(54,194,64,204)(55,195,65,205)(56,196,66,206)(57,197,75,215)(58,198,76,216)(59,199,77,217)(60,200,71,211)(61,201,72,212)(62,202,73,213)(63,203,74,214)(78,186,95,175)(79,187,96,169)(80,188,97,170)(81,189,98,171)(82,183,92,172)(83,184,93,173)(84,185,94,174)(85,180,103,163)(86,181,104,164)(87,182,105,165)(88,176,99,166)(89,177,100,167)(90,178,101,168)(91,179,102,162)(106,158,123,147)(107,159,124,141)(108,160,125,142)(109,161,126,143)(110,155,120,144)(111,156,121,145)(112,157,122,146)(113,152,131,135)(114,153,132,136)(115,154,133,137)(116,148,127,138)(117,149,128,139)(118,150,129,140)(119,151,130,134), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,110)(9,111)(10,112)(11,106)(12,107)(13,108)(14,109)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,120)(22,117)(23,118)(24,119)(25,113)(26,114)(27,115)(28,116)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,186)(51,187)(52,188)(53,189)(54,183)(55,184)(56,185)(57,180)(58,181)(59,182)(60,176)(61,177)(62,178)(63,179)(64,172)(65,173)(66,174)(67,175)(68,169)(69,170)(70,171)(71,166)(72,167)(73,168)(74,162)(75,163)(76,164)(77,165)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(127,222)(128,223)(129,224)(130,218)(131,219)(132,220)(133,221)>;
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), (1,63,34,67)(2,57,35,68)(3,58,29,69)(4,59,30,70)(5,60,31,64)(6,61,32,65)(7,62,33,66)(8,200,222,204)(9,201,223,205)(10,202,224,206)(11,203,218,207)(12,197,219,208)(13,198,220,209)(14,199,221,210)(15,212,22,195)(16,213,23,196)(17,214,24,190)(18,215,25,191)(19,216,26,192)(20,217,27,193)(21,211,28,194)(36,54,43,71)(37,55,44,72)(38,56,45,73)(39,50,46,74)(40,51,47,75)(41,52,48,76)(42,53,49,77)(78,123,102,119)(79,124,103,113)(80,125,104,114)(81,126,105,115)(82,120,99,116)(83,121,100,117)(84,122,101,118)(85,131,96,107)(86,132,97,108)(87,133,98,109)(88,127,92,110)(89,128,93,111)(90,129,94,112)(91,130,95,106)(134,175,158,179)(135,169,159,180)(136,170,160,181)(137,171,161,182)(138,172,155,176)(139,173,156,177)(140,174,157,178)(141,163,152,187)(142,164,153,188)(143,165,154,189)(144,166,148,183)(145,167,149,184)(146,168,150,185)(147,162,151,186), (50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,68)(58,69)(59,70)(60,64)(61,65)(62,66)(63,67)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,124)(114,125)(115,126)(116,120)(117,121)(118,122)(119,123)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,152)(142,153)(143,154)(144,148)(145,149)(146,150)(147,151)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,208)(198,209)(199,210)(200,204)(201,205)(202,206)(203,207), (1,11,46,17)(2,12,47,18)(3,13,48,19)(4,14,49,20)(5,8,43,21)(6,9,44,15)(7,10,45,16)(22,32,223,37)(23,33,224,38)(24,34,218,39)(25,35,219,40)(26,29,220,41)(27,30,221,42)(28,31,222,36)(50,190,67,207)(51,191,68,208)(52,192,69,209)(53,193,70,210)(54,194,64,204)(55,195,65,205)(56,196,66,206)(57,197,75,215)(58,198,76,216)(59,199,77,217)(60,200,71,211)(61,201,72,212)(62,202,73,213)(63,203,74,214)(78,186,95,175)(79,187,96,169)(80,188,97,170)(81,189,98,171)(82,183,92,172)(83,184,93,173)(84,185,94,174)(85,180,103,163)(86,181,104,164)(87,182,105,165)(88,176,99,166)(89,177,100,167)(90,178,101,168)(91,179,102,162)(106,158,123,147)(107,159,124,141)(108,160,125,142)(109,161,126,143)(110,155,120,144)(111,156,121,145)(112,157,122,146)(113,152,131,135)(114,153,132,136)(115,154,133,137)(116,148,127,138)(117,149,128,139)(118,150,129,140)(119,151,130,134), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,110)(9,111)(10,112)(11,106)(12,107)(13,108)(14,109)(15,121)(16,122)(17,123)(18,124)(19,125)(20,126)(21,120)(22,117)(23,118)(24,119)(25,113)(26,114)(27,115)(28,116)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,144)(44,145)(45,146)(46,147)(47,141)(48,142)(49,143)(50,186)(51,187)(52,188)(53,189)(54,183)(55,184)(56,185)(57,180)(58,181)(59,182)(60,176)(61,177)(62,178)(63,179)(64,172)(65,173)(66,174)(67,175)(68,169)(69,170)(70,171)(71,166)(72,167)(73,168)(74,162)(75,163)(76,164)(77,165)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(127,222)(128,223)(129,224)(130,218)(131,219)(132,220)(133,221) );
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)], [(1,63,34,67),(2,57,35,68),(3,58,29,69),(4,59,30,70),(5,60,31,64),(6,61,32,65),(7,62,33,66),(8,200,222,204),(9,201,223,205),(10,202,224,206),(11,203,218,207),(12,197,219,208),(13,198,220,209),(14,199,221,210),(15,212,22,195),(16,213,23,196),(17,214,24,190),(18,215,25,191),(19,216,26,192),(20,217,27,193),(21,211,28,194),(36,54,43,71),(37,55,44,72),(38,56,45,73),(39,50,46,74),(40,51,47,75),(41,52,48,76),(42,53,49,77),(78,123,102,119),(79,124,103,113),(80,125,104,114),(81,126,105,115),(82,120,99,116),(83,121,100,117),(84,122,101,118),(85,131,96,107),(86,132,97,108),(87,133,98,109),(88,127,92,110),(89,128,93,111),(90,129,94,112),(91,130,95,106),(134,175,158,179),(135,169,159,180),(136,170,160,181),(137,171,161,182),(138,172,155,176),(139,173,156,177),(140,174,157,178),(141,163,152,187),(142,164,153,188),(143,165,154,189),(144,166,148,183),(145,167,149,184),(146,168,150,185),(147,162,151,186)], [(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,68),(58,69),(59,70),(60,64),(61,65),(62,66),(63,67),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,124),(114,125),(115,126),(116,120),(117,121),(118,122),(119,123),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,152),(142,153),(143,154),(144,148),(145,149),(146,150),(147,151),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,208),(198,209),(199,210),(200,204),(201,205),(202,206),(203,207)], [(1,11,46,17),(2,12,47,18),(3,13,48,19),(4,14,49,20),(5,8,43,21),(6,9,44,15),(7,10,45,16),(22,32,223,37),(23,33,224,38),(24,34,218,39),(25,35,219,40),(26,29,220,41),(27,30,221,42),(28,31,222,36),(50,190,67,207),(51,191,68,208),(52,192,69,209),(53,193,70,210),(54,194,64,204),(55,195,65,205),(56,196,66,206),(57,197,75,215),(58,198,76,216),(59,199,77,217),(60,200,71,211),(61,201,72,212),(62,202,73,213),(63,203,74,214),(78,186,95,175),(79,187,96,169),(80,188,97,170),(81,189,98,171),(82,183,92,172),(83,184,93,173),(84,185,94,174),(85,180,103,163),(86,181,104,164),(87,182,105,165),(88,176,99,166),(89,177,100,167),(90,178,101,168),(91,179,102,162),(106,158,123,147),(107,159,124,141),(108,160,125,142),(109,161,126,143),(110,155,120,144),(111,156,121,145),(112,157,122,146),(113,152,131,135),(114,153,132,136),(115,154,133,137),(116,148,127,138),(117,149,128,139),(118,150,129,140),(119,151,130,134)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,110),(9,111),(10,112),(11,106),(12,107),(13,108),(14,109),(15,121),(16,122),(17,123),(18,124),(19,125),(20,126),(21,120),(22,117),(23,118),(24,119),(25,113),(26,114),(27,115),(28,116),(29,136),(30,137),(31,138),(32,139),(33,140),(34,134),(35,135),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,144),(44,145),(45,146),(46,147),(47,141),(48,142),(49,143),(50,186),(51,187),(52,188),(53,189),(54,183),(55,184),(56,185),(57,180),(58,181),(59,182),(60,176),(61,177),(62,178),(63,179),(64,172),(65,173),(66,174),(67,175),(68,169),(69,170),(70,171),(71,166),(72,167),(73,168),(74,162),(75,163),(76,164),(77,165),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(127,222),(128,223),(129,224),(130,218),(131,219),(132,220),(133,221)]])
175 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4H | 4I | ··· | 4O | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AP | 14AQ | ··· | 14BB | 28A | ··· | 28AV | 28AW | ··· | 28CL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
175 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 | 2- 1+4 | C7×2- 1+4 |
| kernel | C7×D4⋊6D4 | C14×C4⋊C4 | D4×C28 | C7×C4⋊D4 | C7×C22⋊Q8 | C7×C22.D4 | C7×C4⋊Q8 | C14×C4○D4 | D4⋊6D4 | C2×C4⋊C4 | C4×D4 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4⋊Q8 | C2×C4○D4 | C7×D4 | C28 | D4 | C4 | C14 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 2 | 6 | 12 | 12 | 12 | 12 | 24 | 6 | 12 | 4 | 4 | 24 | 24 | 1 | 6 |
Matrix representation of C7×D4⋊6D4 ►in GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 25 | 3 |
| 0 | 0 | 4 | 4 |
| 0 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 25 | 18 |
| 0 | 0 | 4 | 4 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,28,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,28,0,0,0,0,25,4,0,0,3,4],[0,17,0,0,12,0,0,0,0,0,25,4,0,0,18,4] >;
C7×D4⋊6D4 in GAP, Magma, Sage, TeX
C_7\times D_4\rtimes_6D_4
% in TeX
G:=Group("C7xD4:6D4"); // GroupNames label
G:=SmallGroup(448,1330);
// by ID
G=gap.SmallGroup(448,1330);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,4790,604,1690]);
// 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,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,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations