D7xC4^2.C2 - GroupNames

G = D₇×C₄².C₂ order 448 = 2⁶·7

Direct product of D₇ and C₄².C₂

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

Aliases: D₇×C₄².C₂, C₄².₂₃₅D₁₄, (C₄×D₇).₇Q₈, C₄.₃₇(Q₈×D₇), C₂₈.₄₈(C₂×Q₈), C₄⋊C₄.₂₀₃D₁₄, D₁₄.₁₅(C₂×Q₈), Dic₇.₅(C₂×Q₈), (D₇×C₄²).₇C₂, Dic₇.Q₈⋊₃₁C₂, (C₂×C₂₈).₈₅C₂³, C₂₈.₃Q₈⋊₃₂C₂, C₂₈.₆Q₈⋊₂₁C₂, D₁₄.₄₁(C₄○D₄), C₁₄.₄₀(C₂²×Q₈), (C₄×C₂₈).₁₉₁C₂², (C₂×C₁₄).₂₃₁C₂⁴, Dic₇⋊C₄.₇₁C₂², C₄⋊Dic₇.₂₃₈C₂², C₂².₂₅₂(C₂³×D₇), (C₂×Dic₇).₂₅₇C₂³, (C₄×Dic₇).₂₉₇C₂², (C₂²×D₇).₂₅₇C₂³, C₂.₂₃(C₂×Q₈×D₇), C₇⋊₄(C₂×C₄².C₂), (D₇×C₄⋊C₄).₁₀C₂, C₂.₈₃(D₇×C₄○D₄), (C₇×C₄².C₂)⋊₄C₂, C₁₄.₁₉₄(C₂×C₄○D₄), (C₂×C₄×D₇).₂₄₈C₂², (C₂×C₄).₇₆(C₂²×D₇), (C₇×C₄⋊C₄).₁₈₆C₂², SmallGroup(448,1140)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₇×C₄².C₂

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄² — D₇×C₄².C₂

Lower central

C₇ — C₂×C₁₄ — D₇×C₄².C₂

Upper central

C₁ — C₂² — C₄².C₂

Generators and relations for D₇×C₄².C₂
G = < a,b,c,d,e | a⁷=b²=c⁴=d⁴=1, e²=d², bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece^-1=cd², ede^-1=c²d >

Subgroups: 892 in 226 conjugacy classes, 111 normal (19 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, D₇, C₁₄, C₁₄, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂²×C₄, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₄², C₂×C₄⋊C₄, C₄².C₂, C₄².C₂, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂²×D₇, C₂×C₄².C₂, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄×C₂₈, C₇×C₄⋊C₄, C₂×C₄×D₇, C₂×C₄×D₇, C₂₈.₆Q₈, D₇×C₄², Dic₇.Q₈, C₂₈.₃Q₈, D₇×C₄⋊C₄, C₇×C₄².C₂, D₇×C₄².C₂
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₄○D₄, C₂⁴, D₁₄, C₄².C₂, C₂²×Q₈, C₂×C₄○D₄, C₂²×D₇, C₂×C₄².C₂, Q₈×D₇, C₂³×D₇, C₂×Q₈×D₇, D₇×C₄○D₄, D₇×C₄².C₂

Smallest permutation representation of D₇×C₄².C₂
►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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)(36 38)(39 42)(40 41)(43 45)(46 49)(47 48)(50 52)(53 56)(54 55)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 87)(88 91)(89 90)(92 94)(95 98)(96 97)(99 101)(102 105)(103 104)(106 108)(109 112)(110 111)(113 115)(116 119)(117 118)(120 122)(123 126)(124 125)(127 129)(130 133)(131 132)(134 136)(137 140)(138 139)(141 143)(144 147)(145 146)(148 150)(151 154)(152 153)(155 157)(158 161)(159 160)(162 164)(165 168)(166 167)(169 171)(172 175)(173 174)(176 178)(179 182)(180 181)(183 185)(186 189)(187 188)(190 192)(193 196)(194 195)(197 199)(200 203)(201 202)(204 206)(207 210)(208 209)(211 213)(214 217)(215 216)(218 220)(221 224)(222 223)

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

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

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4F	4G	4H	4I	4J	4K	···	4P	4Q	4R	4S	4T	7A	7B	7C	14A	···	14I	28A	···	28R	28S	···	28AD
order	1	2	2	2	2	2	2	2	4	···	4	4	4	4	4	4	···	4	4	4	4	4	7	7	7	14	···	14	28	···	28	28	···	28
size	1	1	1	1	7	7	7	7	2	···	2	4	4	4	4	14	···	14	28	28	28	28	2	2	2	2	···	2	4	···	4	8	···	8

70 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	2	4	4
type	+	+	+	+	+	+	+	-	+		+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₇	C₄○D₄	D₁₄	D₁₄	Q₈×D₇	D₇×C₄○D₄
kernel	D₇×C₄².C₂	C₂₈.₆Q₈	D₇×C₄²	Dic₇.Q₈	C₂₈.₃Q₈	D₇×C₄⋊C₄	C₇×C₄².C₂	C₄×D₇	C₄².C₂	D₁₄	C₄²	C₄⋊C₄	C₄	C₂
# reps	1	1	1	4	2	6	1	4	3	8	3	18	6	12

Matrix representation of D₇×C₄².C₂ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	18	1	0	0
0	0	28	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	25	0	0
0	0	11	25	0	0
0	0	0	0	1	0
0	0	0	0	0	1

14	16	0	0	0	0
15	15	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	8	15
0	0	0	0	15	21

17	0	0	0	0	0
0	17	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	21	14
0	0	0	0	14	8

14	12	0	0	0	0
15	15	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	14	8
0	0	0	0	8	15

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,28,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,11,0,0,0,0,25,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[14,15,0,0,0,0,16,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,15,0,0,0,0,15,21],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,14,0,0,0,0,14,8],[14,15,0,0,0,0,12,15,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,14,8,0,0,0,0,8,15] >;

D₇×C₄².C₂ in GAP, Magma, Sage, TeX

D_7\times C_4^2.C_2

% in TeX

G:=Group("D7xC4^2.C2");

// GroupNames label

G:=SmallGroup(448,1140);

// by ID

G=gap.SmallGroup(448,1140);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,346,297,136,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^2=c^4=d^4=1,e^2=d^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c*d^2,e*d*e^-1=c^2*d>;

// generators/relations

G = D7×C42.C2 order 448 = 26·7

Direct product of D7 and C42.C2

G = D₇×C₄².C₂ order 448 = 2⁶·7

Direct product of D₇ and C₄².C₂