Copied to
clipboard

?

G = C2×Dic7⋊D4  order 448 = 26·7

Direct product of C2 and Dic7⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic7⋊D4, C24.59D14, (C2×D4)⋊38D14, Dic7⋊8(C2×D4), (C22×D4)⋊8D7, C14⋊5(C4⋊D4), (C2×Dic7)⋊22D4, (C22×C14)⋊12D4, C23⋊5(C7⋊D4), D14⋊C4⋊72C22, (D4×C14)⋊57C22, (C23×Dic7)⋊9C2, C22.148(D4×D7), (C2×C14).297C24, (C2×C28).643C23, Dic7⋊C4⋊74C22, C14.144(C22×D4), (C22×C4).271D14, C23.D7⋊63C22, (C23×C14).77C22, (C23×D7).76C22, C23.338(C22×D7), C22.310(C23×D7), C22.80(D4⋊2D7), (C22×C28).438C22, (C22×C14).231C23, (C2×Dic7).284C23, (C22×Dic7)⋊49C22, (C22×D7).128C23, C7⋊6(C2×C4⋊D4), (D4×C2×C14)⋊16C2, (C2×C14)⋊9(C2×D4), C2.104(C2×D4×D7), C22⋊1(C2×C7⋊D4), (C2×D14⋊C4)⋊42C2, (C2×Dic7⋊C4)⋊48C2, C14.106(C2×C4○D4), C2.70(C2×D4⋊2D7), (C22×C7⋊D4)⋊15C2, (C2×C7⋊D4)⋊46C22, (C2×C23.D7)⋊29C2, C2.17(C22×C7⋊D4), (C2×C4).237(C22×D7), (C2×C14).178(C4○D4), SmallGroup(448,1255)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C14 — C2×Dic7⋊D4
Chief series
C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — C22×C7⋊D4 — C2×Dic7⋊D4
Lower central
C7 — C2×C14 — C2×Dic7⋊D4

Subgroups: 1876 in 426 conjugacy classes, 135 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×10], C22, C22 [×10], C22 [×32], C7, C2×C4 [×2], C2×C4 [×24], D4 [×24], C23, C23 [×8], C23 [×18], D7 [×2], C14 [×3], C14 [×4], C14 [×6], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24, Dic7 [×4], Dic7 [×4], C28 [×2], D14 [×10], C2×C14, C2×C14 [×10], C2×C14 [×22], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4, C22×D4 [×2], C2×Dic7 [×10], C2×Dic7 [×12], C7⋊D4 [×16], C2×C28 [×2], C2×C28 [×2], C7×D4 [×8], C22×D7 [×2], C22×D7 [×6], C22×C14, C22×C14 [×8], C22×C14 [×10], C2×C4⋊D4, Dic7⋊C4 [×4], D14⋊C4 [×4], C23.D7 [×4], C22×Dic7 [×3], C22×Dic7 [×4], C22×Dic7 [×4], C2×C7⋊D4 [×8], C2×C7⋊D4 [×8], C22×C28, D4×C14 [×4], D4×C14 [×4], C23×D7, C23×C14 [×2], C2×Dic7⋊C4, C2×D14⋊C4, Dic7⋊D4 [×8], C2×C23.D7, C23×Dic7, C22×C7⋊D4 [×2], D4×C2×C14, C2×Dic7⋊D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D7, C2×D4 [×12], C4○D4 [×2], C24, D14 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C7⋊D4 [×4], C22×D7 [×7], C2×C4⋊D4, D4×D7 [×2], D4⋊2D7 [×2], C2×C7⋊D4 [×6], C23×D7, Dic7⋊D4 [×4], C2×D4×D7, C2×D4⋊2D7, C22×C7⋊D4, C2×Dic7⋊D4

Generators and relations
 G = < a,b,c,d,e | a2=b14=d4=e2=1, c2=b7, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b7c, ce=ec, ede=d-1 >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽29)

2800000
0280000
0028000
0002800
0000280
0000028
,
21280000
2110000
0026100
0028000
0000280
0000028
,
2720000
1320000
0001200
0012000
00001228
0000017
,
2800000
0280000
0051600
00132400
00002817
000051
,
100000
010000
001000
000100
0000112
0000028

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[21,2,0,0,0,0,28,11,0,0,0,0,0,0,26,28,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[27,13,0,0,0,0,2,2,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,28,17],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,5,13,0,0,0,0,16,24,0,0,0,0,0,0,28,5,0,0,0,0,17,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,12,28] >;
 

88 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C···4J4K4L7A7B7C14A···14U14V···14AS28A···28L
order12···222222222444···44477714···1414···1428···28
size11···122224428284414···1428282222···24···44···4

88 irreducible representations

dim111111112222222244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2D4D4D7C4○D4D14D14D14C7⋊D4D4×D7D4⋊2D7
kernelC2×Dic7⋊D4C2×Dic7⋊C4C2×D14⋊C4Dic7⋊D4C2×C23.D7C23×Dic7C22×C7⋊D4D4×C2×C14C2×Dic7C22×C14C22×D4C2×C14C22×C4C2×D4C24C23C22C22
# reps11181121443431262466

In GAP, Magma, Sage, TeX 

C_2\times Dic_7\rtimes D_4
 
% in TeX
 
G:=Group("C2xDic7:D4");
 
// GroupNames label
 
G:=SmallGroup(448,1255);
 
// by ID
 
G=gap.SmallGroup(448,1255);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,758,675,297,18822]);
 
// 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,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^7*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀