(C2xD4).D14 - GroupNames

G = (C₂×D₄).D₁₄ order 448 = 2⁶·7

36^th non-split extension by C₂×D₄ of D₁₄ acting via D₁₄/C₇=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₄⋊C₄.₅₆D₁₄, C₄⋊D₄.₄D₇, (C₂×D₄).₃₆D₁₄, (C₂×C₂₈).₂₆₁D₄, C₁₄.₉₅(C₄○D₈), D₄⋊Dic₇⋊₁₄C₂, C₂₈.₅₅D₄⋊₉C₂, C₂₈.Q₈⋊₃₅C₂, C₄.Dic₁₄⋊₃₄C₂, (C₂²×C₁₄).₈₁D₄, C₂₈.₁₈₂(C₄○D₄), C₄.₉₂(D₄⋊₂D₇), C₁₄.₈₉(C₈⋊C₂²), (C₂×C₂₈).₃₅₄C₂³, (D₄×C₁₄).₅₂C₂², (C₂²×C₄).₁₁₈D₁₄, C₂³.₂₂(C₇⋊D₄), C₇⋊₇(C₂³.₁₉D₄), C₄⋊Dic₇.₃₃₆C₂², C₂.₁₀(D₄.D₁₄), C₂.₁₄(D₄.₈D₁₄), C₂³.₂₁D₁₄⋊₁₅C₂, (C₂²×C₂₈).₁₅₈C₂², C₁₄.₇₉(C₂².D₄), C₂.₁₃(C₂³.₁₈D₁₄), (C₇×C₄⋊D₄).₃C₂, (C₂×C₁₄).₄₈₅(C₂×D₄), (C₂×C₇⋊C₈).₁₀₇C₂², (C₂×C₄).₁₇₀(C₇⋊D₄), (C₇×C₄⋊C₄).₁₀₃C₂², (C₂×C₄).₄₅₄(C₂²×D₇), C₂².₁₆₀(C₂×C₇⋊D₄), SmallGroup(448,569)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — (C₂×D₄).D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₂×C₂₈ — C₄⋊Dic₇ — C₂³.₂₁D₁₄ — (C₂×D₄).D₁₄

Lower central

C₇ — C₁₄ — C₂×C₂₈ — (C₂×D₄).D₁₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₄⋊D₄

Generators and relations for (C₂×D₄).D₁₄
G = < a,b,c,d,e | a²=b⁴=c²=1, d¹⁴=b², e²=a, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe^-1=b^-1, bd=db, dcd^-1=ab²c, ece^-1=b^-1c, ede^-1=d¹³ >

Subgroups: 396 in 106 conjugacy classes, 39 normal (all characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂×D₄, C₂×D₄, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂²⋊C₈, D₄⋊C₄, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₄⋊D₄, C₇⋊C₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₂²×C₁₄, C₂³.₁₉D₄, C₂×C₇⋊C₈, C₄×Dic₇, C₄⋊Dic₇, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₂²×C₂₈, D₄×C₁₄, D₄×C₁₄, C₂₈.Q₈, C₄.Dic₁₄, C₂₈.₅₅D₄, D₄⋊Dic₇, C₂³.₂₁D₁₄, C₇×C₄⋊D₄, (C₂×D₄).D₁₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, D₁₄, C₂².D₄, C₄○D₈, C₈⋊C₂², C₇⋊D₄, C₂²×D₇, C₂³.₁₉D₄, D₄⋊₂D₇, C₂×C₇⋊D₄, D₄.D₁₄, C₂³.₁₈D₁₄, D₄.₈D₁₄, (C₂×D₄).D₁₄

Smallest permutation representation of (C₂×D₄).D₁₄
►On 224 points

Generators in S₂₂₄

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

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

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

(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)

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

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

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

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

61 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	14J	···	14O	14P	···	14U	28A	···	28L	28M	···	28R
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	1	1	4	8	2	2	2	2	8	28	28	28	28	2	2	2	28	28	28	28	2	···	2	4	···	4	8	···	8	4	···	4	8	···	8

61 irreducible representations

dim	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	D₄	D₄	D₇	C₄○D₄	D₁₄	D₁₄	D₁₄	C₄○D₈	C₇⋊D₄	C₇⋊D₄	C₈⋊C₂²	D₄⋊₂D₇	D₄.D₁₄	D₄.₈D₁₄
kernel	(C₂×D₄).D₁₄	C₂₈.Q₈	C₄.Dic₁₄	C₂₈.₅₅D₄	D₄⋊Dic₇	C₂³.₂₁D₁₄	C₇×C₄⋊D₄	C₂×C₂₈	C₂²×C₁₄	C₄⋊D₄	C₂₈	C₄⋊C₄	C₂²×C₄	C₂×D₄	C₁₄	C₂×C₄	C₂³	C₁₄	C₄	C₂	C₂
# reps	1	1	1	1	2	1	1	1	1	3	4	3	3	3	4	6	6	1	6	6	6

Matrix representation of (C₂×D₄).D₁₄ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

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	98	0
0	0	0	0	57	15

1	0	0	0	0	0
0	1	0	0	0	0
0	0	87	59	0	0
0	0	69	26	0	0
0	0	0	0	40	27
0	0	0	0	104	73

30	0	0	0	0	0
1	49	0	0	0	0
0	0	112	0	0	0
0	0	47	1	0	0
0	0	0	0	15	0
0	0	0	0	0	15

59	104	0	0	0	0
10	54	0	0	0	0
0	0	15	0	0	0
0	0	0	15	0	0
0	0	0	0	48	55
0	0	0	0	30	65

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,98,57,0,0,0,0,0,15],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,87,69,0,0,0,0,59,26,0,0,0,0,0,0,40,104,0,0,0,0,27,73],[30,1,0,0,0,0,0,49,0,0,0,0,0,0,112,47,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[59,10,0,0,0,0,104,54,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,48,30,0,0,0,0,55,65] >;

(C₂×D₄).D₁₄ in GAP, Magma, Sage, TeX

(C_2\times D_4).D_{14}

% in TeX

G:=Group("(C2xD4).D14");

// GroupNames label

G:=SmallGroup(448,569);

// by ID

G=gap.SmallGroup(448,569);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,112,232,254,219,1123,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^14=b^2,e^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a*b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^13>;

// generators/relations

G = (C2×D4).D14 order 448 = 26·7

36th non-split extension by C2×D4 of D14 acting via D14/C7=C22

G = (C₂×D₄).D₁₄ order 448 = 2⁶·7

36^th non-split extension by C₂×D₄ of D₁₄ acting via D₁₄/C₇=C₂²