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G = (C2×Dic7)⋊3D4  order 448 = 26·7

1st semidirect product of C2×Dic7 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic7)⋊3D4, (C2×C28).30D4, (C2×C4).19D28, C14.5C22≀C2, (C22×D7).7D4, C2.8(C4⋊D28), C14.6(C4⋊D4), (C22×D28).2C2, C22.81(C2×D28), (C22×C4).71D14, C22.156(D4×D7), C2.9(D14⋊D4), C2.8(C22⋊D28), C2.6(C4.D28), C14.2(C4.4D4), C2.C42⋊11D7, C7⋊1(C23.10D4), (C23×D7).4C22, C2.9(D14.5D4), C22.89(C4○D28), (C22×C28).16C22, C23.360(C22×D7), (C22×C14).297C23, C22.45(Q8⋊2D7), C14.40(C22.D4), (C22×Dic7).19C22, (C2×D14⋊C4)⋊1C2, (C2×Dic7⋊C4)⋊19C2, (C2×C14).205(C2×D4), (C2×C14).59(C4○D4), (C7×C2.C42)⋊9C2, SmallGroup(448,206)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C14 — (C2×Dic7)⋊3D4
Chief series
C1 — C7 — C14 — C2×C14 — C22×C14 — C23×D7 — C2×D14⋊C4 — (C2×Dic7)⋊3D4
Lower central
C7 — C22×C14 — (C2×Dic7)⋊3D4
Upper central
C1 — C23 — C2.C42

Generators and relations for (C2×Dic7)⋊3D4
 G = < a,b,c,d,e | a2=b14=d4=e2=1, c2=b7, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, dcd-1=b7c, ede=d-1 >

Subgroups: 1532 in 238 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, D7, C14, C14, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, Dic7, C28, D14, C2×C14, C2×C14, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D28, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C22×D7, C22×C14, C23.10D4, Dic7⋊C4, D14⋊C4, C2×D28, C22×Dic7, C22×C28, C22×C28, C23×D7, C7×C2.C42, C2×Dic7⋊C4, C2×D14⋊C4, C22×D28, (C2×Dic7)⋊3D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C22≀C2, C4⋊D4, C22.D4, C4.4D4, D28, C22×D7, C23.10D4, C2×D28, C4○D28, D4×D7, Q8⋊2D7, C4.D28, C22⋊D28, D14⋊D4, D14.5D4, C4⋊D28, (C2×Dic7)⋊3D4

Smallest permutation representation of (C2×Dic7)⋊3D4
►On 224 points
Generators in S224
(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 203)(16 204)(17 205)(18 206)(19 207)(20 208)(21 209)(22 210)(23 197)(24 198)(25 199)(26 200)(27 201)(28 202)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(57 161)(58 162)(59 163)(60 164)(61 165)(62 166)(63 167)(64 168)(65 155)(66 156)(67 157)(68 158)(69 159)(70 160)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 127)(99 174)(100 175)(101 176)(102 177)(103 178)(104 179)(105 180)(106 181)(107 182)(108 169)(109 170)(110 171)(111 172)(112 173)(113 183)(114 184)(115 185)(116 186)(117 187)(118 188)(119 189)(120 190)(121 191)(122 192)(123 193)(124 194)(125 195)(126 196)(141 212)(142 213)(143 214)(144 215)(145 216)(146 217)(147 218)(148 219)(149 220)(150 221)(151 222)(152 223)(153 224)(154 211)

(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)

(1 220 8 213)(2 219 9 212)(3 218 10 211)(4 217 11 224)(5 216 12 223)(6 215 13 222)(7 214 14 221)(15 176 22 169)(16 175 23 182)(17 174 24 181)(18 173 25 180)(19 172 26 179)(20 171 27 178)(21 170 28 177)(29 162 36 155)(30 161 37 168)(31 160 38 167)(32 159 39 166)(33 158 40 165)(34 157 41 164)(35 156 42 163)(43 64 50 57)(44 63 51 70)(45 62 52 69)(46 61 53 68)(47 60 54 67)(48 59 55 66)(49 58 56 65)(71 143 78 150)(72 142 79 149)(73 141 80 148)(74 154 81 147)(75 153 82 146)(76 152 83 145)(77 151 84 144)(85 113 92 120)(86 126 93 119)(87 125 94 118)(88 124 95 117)(89 123 96 116)(90 122 97 115)(91 121 98 114)(99 198 106 205)(100 197 107 204)(101 210 108 203)(102 209 109 202)(103 208 110 201)(104 207 111 200)(105 206 112 199)(127 184 134 191)(128 183 135 190)(129 196 136 189)(130 195 137 188)(131 194 138 187)(132 193 139 186)(133 192 140 185)

(1 122 42 104)(2 123 29 105)(3 124 30 106)(4 125 31 107)(5 126 32 108)(6 113 33 109)(7 114 34 110)(8 115 35 111)(9 116 36 112)(10 117 37 99)(11 118 38 100)(12 119 39 101)(13 120 40 102)(14 121 41 103)(15 152 136 62)(16 153 137 63)(17 154 138 64)(18 141 139 65)(19 142 140 66)(20 143 127 67)(21 144 128 68)(22 145 129 69)(23 146 130 70)(24 147 131 57)(25 148 132 58)(26 149 133 59)(27 150 134 60)(28 151 135 61)(43 174 74 187)(44 175 75 188)(45 176 76 189)(46 177 77 190)(47 178 78 191)(48 179 79 192)(49 180 80 193)(50 181 81 194)(51 182 82 195)(52 169 83 196)(53 170 84 183)(54 171 71 184)(55 172 72 185)(56 173 73 186)(85 158 209 215)(86 159 210 216)(87 160 197 217)(88 161 198 218)(89 162 199 219)(90 163 200 220)(91 164 201 221)(92 165 202 222)(93 166 203 223)(94 167 204 224)(95 168 205 211)(96 155 206 212)(97 156 207 213)(98 157 208 214)

(1 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 42)(9 41)(10 40)(11 39)(12 38)(13 37)(14 36)(15 204)(16 203)(17 202)(18 201)(19 200)(20 199)(21 198)(22 197)(23 210)(24 209)(25 208)(26 207)(27 206)(28 205)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 84)(51 83)(52 82)(53 81)(54 80)(55 79)(56 78)(57 215)(58 214)(59 213)(60 212)(61 211)(62 224)(63 223)(64 222)(65 221)(66 220)(67 219)(68 218)(69 217)(70 216)(85 131)(86 130)(87 129)(88 128)(89 127)(90 140)(91 139)(92 138)(93 137)(94 136)(95 135)(96 134)(97 133)(98 132)(99 102)(100 101)(103 112)(104 111)(105 110)(106 109)(107 108)(113 124)(114 123)(115 122)(116 121)(117 120)(118 119)(125 126)(141 164)(142 163)(143 162)(144 161)(145 160)(146 159)(147 158)(148 157)(149 156)(150 155)(151 168)(152 167)(153 166)(154 165)(169 182)(170 181)(171 180)(172 179)(173 178)(174 177)(175 176)(183 194)(184 193)(185 192)(186 191)(187 190)(188 189)(195 196)
 
G:=sub<Sym(224)| (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,203)(16,204)(17,205)(18,206)(19,207)(20,208)(21,209)(22,210)(23,197)(24,198)(25,199)(26,200)(27,201)(28,202)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,127)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,169)(109,170)(110,171)(111,172)(112,173)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,211), (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), (1,220,8,213)(2,219,9,212)(3,218,10,211)(4,217,11,224)(5,216,12,223)(6,215,13,222)(7,214,14,221)(15,176,22,169)(16,175,23,182)(17,174,24,181)(18,173,25,180)(19,172,26,179)(20,171,27,178)(21,170,28,177)(29,162,36,155)(30,161,37,168)(31,160,38,167)(32,159,39,166)(33,158,40,165)(34,157,41,164)(35,156,42,163)(43,64,50,57)(44,63,51,70)(45,62,52,69)(46,61,53,68)(47,60,54,67)(48,59,55,66)(49,58,56,65)(71,143,78,150)(72,142,79,149)(73,141,80,148)(74,154,81,147)(75,153,82,146)(76,152,83,145)(77,151,84,144)(85,113,92,120)(86,126,93,119)(87,125,94,118)(88,124,95,117)(89,123,96,116)(90,122,97,115)(91,121,98,114)(99,198,106,205)(100,197,107,204)(101,210,108,203)(102,209,109,202)(103,208,110,201)(104,207,111,200)(105,206,112,199)(127,184,134,191)(128,183,135,190)(129,196,136,189)(130,195,137,188)(131,194,138,187)(132,193,139,186)(133,192,140,185), (1,122,42,104)(2,123,29,105)(3,124,30,106)(4,125,31,107)(5,126,32,108)(6,113,33,109)(7,114,34,110)(8,115,35,111)(9,116,36,112)(10,117,37,99)(11,118,38,100)(12,119,39,101)(13,120,40,102)(14,121,41,103)(15,152,136,62)(16,153,137,63)(17,154,138,64)(18,141,139,65)(19,142,140,66)(20,143,127,67)(21,144,128,68)(22,145,129,69)(23,146,130,70)(24,147,131,57)(25,148,132,58)(26,149,133,59)(27,150,134,60)(28,151,135,61)(43,174,74,187)(44,175,75,188)(45,176,76,189)(46,177,77,190)(47,178,78,191)(48,179,79,192)(49,180,80,193)(50,181,81,194)(51,182,82,195)(52,169,83,196)(53,170,84,183)(54,171,71,184)(55,172,72,185)(56,173,73,186)(85,158,209,215)(86,159,210,216)(87,160,197,217)(88,161,198,218)(89,162,199,219)(90,163,200,220)(91,164,201,221)(92,165,202,222)(93,166,203,223)(94,167,204,224)(95,168,205,211)(96,155,206,212)(97,156,207,213)(98,157,208,214), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,204)(16,203)(17,202)(18,201)(19,200)(20,199)(21,198)(22,197)(23,210)(24,209)(25,208)(26,207)(27,206)(28,205)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,215)(58,214)(59,213)(60,212)(61,211)(62,224)(63,223)(64,222)(65,221)(66,220)(67,219)(68,218)(69,217)(70,216)(85,131)(86,130)(87,129)(88,128)(89,127)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(125,126)(141,164)(142,163)(143,162)(144,161)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,168)(152,167)(153,166)(154,165)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,194)(184,193)(185,192)(186,191)(187,190)(188,189)(195,196)>;
 
G:=Group( (1,79)(2,80)(3,81)(4,82)(5,83)(6,84)(7,71)(8,72)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,203)(16,204)(17,205)(18,206)(19,207)(20,208)(21,209)(22,210)(23,197)(24,198)(25,199)(26,200)(27,201)(28,202)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(57,161)(58,162)(59,163)(60,164)(61,165)(62,166)(63,167)(64,168)(65,155)(66,156)(67,157)(68,158)(69,159)(70,160)(85,128)(86,129)(87,130)(88,131)(89,132)(90,133)(91,134)(92,135)(93,136)(94,137)(95,138)(96,139)(97,140)(98,127)(99,174)(100,175)(101,176)(102,177)(103,178)(104,179)(105,180)(106,181)(107,182)(108,169)(109,170)(110,171)(111,172)(112,173)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(121,191)(122,192)(123,193)(124,194)(125,195)(126,196)(141,212)(142,213)(143,214)(144,215)(145,216)(146,217)(147,218)(148,219)(149,220)(150,221)(151,222)(152,223)(153,224)(154,211), (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), (1,220,8,213)(2,219,9,212)(3,218,10,211)(4,217,11,224)(5,216,12,223)(6,215,13,222)(7,214,14,221)(15,176,22,169)(16,175,23,182)(17,174,24,181)(18,173,25,180)(19,172,26,179)(20,171,27,178)(21,170,28,177)(29,162,36,155)(30,161,37,168)(31,160,38,167)(32,159,39,166)(33,158,40,165)(34,157,41,164)(35,156,42,163)(43,64,50,57)(44,63,51,70)(45,62,52,69)(46,61,53,68)(47,60,54,67)(48,59,55,66)(49,58,56,65)(71,143,78,150)(72,142,79,149)(73,141,80,148)(74,154,81,147)(75,153,82,146)(76,152,83,145)(77,151,84,144)(85,113,92,120)(86,126,93,119)(87,125,94,118)(88,124,95,117)(89,123,96,116)(90,122,97,115)(91,121,98,114)(99,198,106,205)(100,197,107,204)(101,210,108,203)(102,209,109,202)(103,208,110,201)(104,207,111,200)(105,206,112,199)(127,184,134,191)(128,183,135,190)(129,196,136,189)(130,195,137,188)(131,194,138,187)(132,193,139,186)(133,192,140,185), (1,122,42,104)(2,123,29,105)(3,124,30,106)(4,125,31,107)(5,126,32,108)(6,113,33,109)(7,114,34,110)(8,115,35,111)(9,116,36,112)(10,117,37,99)(11,118,38,100)(12,119,39,101)(13,120,40,102)(14,121,41,103)(15,152,136,62)(16,153,137,63)(17,154,138,64)(18,141,139,65)(19,142,140,66)(20,143,127,67)(21,144,128,68)(22,145,129,69)(23,146,130,70)(24,147,131,57)(25,148,132,58)(26,149,133,59)(27,150,134,60)(28,151,135,61)(43,174,74,187)(44,175,75,188)(45,176,76,189)(46,177,77,190)(47,178,78,191)(48,179,79,192)(49,180,80,193)(50,181,81,194)(51,182,82,195)(52,169,83,196)(53,170,84,183)(54,171,71,184)(55,172,72,185)(56,173,73,186)(85,158,209,215)(86,159,210,216)(87,160,197,217)(88,161,198,218)(89,162,199,219)(90,163,200,220)(91,164,201,221)(92,165,202,222)(93,166,203,223)(94,167,204,224)(95,168,205,211)(96,155,206,212)(97,156,207,213)(98,157,208,214), (1,35)(2,34)(3,33)(4,32)(5,31)(6,30)(7,29)(8,42)(9,41)(10,40)(11,39)(12,38)(13,37)(14,36)(15,204)(16,203)(17,202)(18,201)(19,200)(20,199)(21,198)(22,197)(23,210)(24,209)(25,208)(26,207)(27,206)(28,205)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,84)(51,83)(52,82)(53,81)(54,80)(55,79)(56,78)(57,215)(58,214)(59,213)(60,212)(61,211)(62,224)(63,223)(64,222)(65,221)(66,220)(67,219)(68,218)(69,217)(70,216)(85,131)(86,130)(87,129)(88,128)(89,127)(90,140)(91,139)(92,138)(93,137)(94,136)(95,135)(96,134)(97,133)(98,132)(99,102)(100,101)(103,112)(104,111)(105,110)(106,109)(107,108)(113,124)(114,123)(115,122)(116,121)(117,120)(118,119)(125,126)(141,164)(142,163)(143,162)(144,161)(145,160)(146,159)(147,158)(148,157)(149,156)(150,155)(151,168)(152,167)(153,166)(154,165)(169,182)(170,181)(171,180)(172,179)(173,178)(174,177)(175,176)(183,194)(184,193)(185,192)(186,191)(187,190)(188,189)(195,196) );
 
G=PermutationGroup([[(1,79),(2,80),(3,81),(4,82),(5,83),(6,84),(7,71),(8,72),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,203),(16,204),(17,205),(18,206),(19,207),(20,208),(21,209),(22,210),(23,197),(24,198),(25,199),(26,200),(27,201),(28,202),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(57,161),(58,162),(59,163),(60,164),(61,165),(62,166),(63,167),(64,168),(65,155),(66,156),(67,157),(68,158),(69,159),(70,160),(85,128),(86,129),(87,130),(88,131),(89,132),(90,133),(91,134),(92,135),(93,136),(94,137),(95,138),(96,139),(97,140),(98,127),(99,174),(100,175),(101,176),(102,177),(103,178),(104,179),(105,180),(106,181),(107,182),(108,169),(109,170),(110,171),(111,172),(112,173),(113,183),(114,184),(115,185),(116,186),(117,187),(118,188),(119,189),(120,190),(121,191),(122,192),(123,193),(124,194),(125,195),(126,196),(141,212),(142,213),(143,214),(144,215),(145,216),(146,217),(147,218),(148,219),(149,220),(150,221),(151,222),(152,223),(153,224),(154,211)], [(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)], [(1,220,8,213),(2,219,9,212),(3,218,10,211),(4,217,11,224),(5,216,12,223),(6,215,13,222),(7,214,14,221),(15,176,22,169),(16,175,23,182),(17,174,24,181),(18,173,25,180),(19,172,26,179),(20,171,27,178),(21,170,28,177),(29,162,36,155),(30,161,37,168),(31,160,38,167),(32,159,39,166),(33,158,40,165),(34,157,41,164),(35,156,42,163),(43,64,50,57),(44,63,51,70),(45,62,52,69),(46,61,53,68),(47,60,54,67),(48,59,55,66),(49,58,56,65),(71,143,78,150),(72,142,79,149),(73,141,80,148),(74,154,81,147),(75,153,82,146),(76,152,83,145),(77,151,84,144),(85,113,92,120),(86,126,93,119),(87,125,94,118),(88,124,95,117),(89,123,96,116),(90,122,97,115),(91,121,98,114),(99,198,106,205),(100,197,107,204),(101,210,108,203),(102,209,109,202),(103,208,110,201),(104,207,111,200),(105,206,112,199),(127,184,134,191),(128,183,135,190),(129,196,136,189),(130,195,137,188),(131,194,138,187),(132,193,139,186),(133,192,140,185)], [(1,122,42,104),(2,123,29,105),(3,124,30,106),(4,125,31,107),(5,126,32,108),(6,113,33,109),(7,114,34,110),(8,115,35,111),(9,116,36,112),(10,117,37,99),(11,118,38,100),(12,119,39,101),(13,120,40,102),(14,121,41,103),(15,152,136,62),(16,153,137,63),(17,154,138,64),(18,141,139,65),(19,142,140,66),(20,143,127,67),(21,144,128,68),(22,145,129,69),(23,146,130,70),(24,147,131,57),(25,148,132,58),(26,149,133,59),(27,150,134,60),(28,151,135,61),(43,174,74,187),(44,175,75,188),(45,176,76,189),(46,177,77,190),(47,178,78,191),(48,179,79,192),(49,180,80,193),(50,181,81,194),(51,182,82,195),(52,169,83,196),(53,170,84,183),(54,171,71,184),(55,172,72,185),(56,173,73,186),(85,158,209,215),(86,159,210,216),(87,160,197,217),(88,161,198,218),(89,162,199,219),(90,163,200,220),(91,164,201,221),(92,165,202,222),(93,166,203,223),(94,167,204,224),(95,168,205,211),(96,155,206,212),(97,156,207,213),(98,157,208,214)], [(1,35),(2,34),(3,33),(4,32),(5,31),(6,30),(7,29),(8,42),(9,41),(10,40),(11,39),(12,38),(13,37),(14,36),(15,204),(16,203),(17,202),(18,201),(19,200),(20,199),(21,198),(22,197),(23,210),(24,209),(25,208),(26,207),(27,206),(28,205),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,84),(51,83),(52,82),(53,81),(54,80),(55,79),(56,78),(57,215),(58,214),(59,213),(60,212),(61,211),(62,224),(63,223),(64,222),(65,221),(66,220),(67,219),(68,218),(69,217),(70,216),(85,131),(86,130),(87,129),(88,128),(89,127),(90,140),(91,139),(92,138),(93,137),(94,136),(95,135),(96,134),(97,133),(98,132),(99,102),(100,101),(103,112),(104,111),(105,110),(106,109),(107,108),(113,124),(114,123),(115,122),(116,121),(117,120),(118,119),(125,126),(141,164),(142,163),(143,162),(144,161),(145,160),(146,159),(147,158),(148,157),(149,156),(150,155),(151,168),(152,167),(153,166),(154,165),(169,182),(170,181),(171,180),(172,179),(173,178),(174,177),(175,176),(183,194),(184,193),(185,192),(186,191),(187,190),(188,189),(195,196)]])
 

82 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4F4G4H4I4J7A7B7C14A···14U28A···28AJ
order12···222224···4444477714···1428···28
size11···1282828284···4282828282222···24···4

82 irreducible representations

dim111112222222244
type+++++++++++++
imageC1C2C2C2C2D4D4D4D7C4○D4D14D28C4○D28D4×D7Q8⋊2D7
kernel(C2×Dic7)⋊3D4C7×C2.C42C2×Dic7⋊C4C2×D14⋊C4C22×D28C2×Dic7C2×C28C22×D7C2.C42C2×C14C22×C4C2×C4C22C22C22
# reps11141224369122493

Matrix representation of (C2×Dic7)⋊3D4 ►in GL6(𝔽29)

100000
010000
0028000
0002800
0000280
0000028
,
810000
27180000
0028000
0002800
0000280
0000028
,
1070000
27190000
00142200
00241500
000019
0000328
,
3110000
7260000
0023300
007600
0000280
0000261
,
4260000
5250000
001000
0042800
0000280
0000261

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[8,27,0,0,0,0,1,18,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[10,27,0,0,0,0,7,19,0,0,0,0,0,0,14,24,0,0,0,0,22,15,0,0,0,0,0,0,1,3,0,0,0,0,9,28],[3,7,0,0,0,0,11,26,0,0,0,0,0,0,23,7,0,0,0,0,3,6,0,0,0,0,0,0,28,26,0,0,0,0,0,1],[4,5,0,0,0,0,26,25,0,0,0,0,0,0,1,4,0,0,0,0,0,28,0,0,0,0,0,0,28,26,0,0,0,0,0,1] >;
 

(C2×Dic7)⋊3D4 in GAP, Magma, Sage, TeX 

(C_2\times {\rm Dic}_7)\rtimes_3D_4
 
% in TeX
 
G:=Group("(C2xDic7):3D4");
 
// GroupNames label
 
G:=SmallGroup(448,206);
 
// by ID
 
G=gap.SmallGroup(448,206);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,64,254,387,268,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^14=d^4=e^2=1,c^2=b^7,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^7*c,e*d*e=d^-1>;
 
// generators/relations
 

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