C2xDic7:D4 - GroupNames

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, C₂².₃₁₀(C₂³×D₇), C₂³.₃₃₈(C₂²×D₇), C₂².₈₀(D₄⋊₂D₇), (C₂²×C₁₄).₂₃₁C₂³, (C₂²×C₂₈).₄₃₈C₂², (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), (C₂×C₇⋊D₄)⋊₄₆C₂², (C₂²×C₇⋊D₄)⋊₁₅C₂, (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

Upper central

C₁ — C₂³ — C₂²×D₄

Generators and relations for C₂×Dic₇⋊D₄
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 >

Subgroups: 1876 in 426 conjugacy classes, 135 normal (31 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₇, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂×C₄⋊D₄, Dic₇⋊C₄, D₁₄⋊C₄, C₂³.D₇, C₂²×Dic₇, C₂²×Dic₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₂×C₇⋊D₄, C₂²×C₂₈, D₄×C₁₄, D₄×C₁₄, C₂³×D₇, C₂³×C₁₄, C₂×Dic₇⋊C₄, C₂×D₁₄⋊C₄, Dic₇⋊D₄, C₂×C₂³.D₇, C₂³×Dic₇, C₂²×C₇⋊D₄, D₄×C₂×C₁₄, C₂×Dic₇⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₄○D₄, C₂⁴, D₁₄, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, C₇⋊D₄, C₂²×D₇, C₂×C₄⋊D₄, D₄×D₇, D₄⋊₂D₇, C₂×C₇⋊D₄, C₂³×D₇, Dic₇⋊D₄, C₂×D₄×D₇, C₂×D₄⋊₂D₇, C₂²×C₇⋊D₄, C₂×Dic₇⋊D₄

Smallest permutation representation of C₂×Dic₇⋊D₄
►On 224 points

Generators in S₂₂₄

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

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

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

(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 71)(42 72)(57 140)(58 127)(59 128)(60 129)(61 130)(62 131)(63 132)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(85 147)(86 148)(87 149)(88 150)(89 151)(90 152)(91 153)(92 154)(93 141)(94 142)(95 143)(96 144)(97 145)(98 146)(183 207)(184 208)(185 209)(186 210)(187 197)(188 198)(189 199)(190 200)(191 201)(192 202)(193 203)(194 204)(195 205)(196 206)

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

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

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

88 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	2M	2N	2O	4A	4B	4C	···	4J	4K	4L	7A	7B	7C	14A	···	14U	14V	···	14AS	28A	···	28L
order	1	2	···	2	2	2	2	2	2	2	2	2	4	4	4	···	4	4	4	7	7	7	14	···	14	14	···	14	28	···	28
size	1	1	···	1	2	2	2	2	4	4	28	28	4	4	14	···	14	28	28	2	2	2	2	···	2	4	···	4	4	···	4

88 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

C₄○D₄

D₁₄

C₇⋊D₄

D₄×D₇

D₄⋊₂D₇

kernel

C₂×Dic₇⋊D₄

C₂×Dic₇⋊C₄

C₂×D₁₄⋊C₄

Dic₇⋊D₄

C₂×C₂³.D₇

C₂³×Dic₇

C₂²×C₇⋊D₄

D₄×C₂×C₁₄

C₂×Dic₇

C₂²×C₁₄

C₂²×D₄

C₂×C₁₄

C₂²×C₄

C₂×D₄

C₂⁴

C₂³

C₂²

# reps

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

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	28	0	0	0	0
2	11	0	0	0	0
0	0	26	1	0	0
0	0	28	0	0	0
0	0	0	0	28	0
0	0	0	0	0	28

27	2	0	0	0	0
13	2	0	0	0	0
0	0	0	12	0	0
0	0	12	0	0	0
0	0	0	0	12	28
0	0	0	0	0	17

28	0	0	0	0	0
0	28	0	0	0	0
0	0	5	16	0	0
0	0	13	24	0	0
0	0	0	0	28	17
0	0	0	0	5	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	12
0	0	0	0	0	28

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] >;

C₂×Dic₇⋊D₄ in GAP, Magma, Sage, TeX

C_2\times {\rm 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

G = C2×Dic7⋊D4 order 448 = 26·7

Direct product of C2 and Dic7⋊D4

G = C₂×Dic₇⋊D₄ order 448 = 2⁶·7

Direct product of C₂ and Dic₇⋊D₄