D7xC4:Q8 - GroupNames

G = D₇×C₄⋊Q₈ order 448 = 2⁶·7

Direct product of D₇ and C₄⋊Q₈

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

Aliases: D₇×C₄⋊Q₈, C₄².₂₃₉D₁₄, C₄⋊₂(Q₈×D₇), (C₄×D₇)⋊₄Q₈, C₂₈⋊₂(C₂×Q₈), C₂₈⋊Q₈⋊₄₃C₂, C₄.₃₆(D₄×D₇), Dic₇⋊₂(C₂×Q₈), (C₄×D₇).₂₅D₄, C₂₈.₆₈(C₂×D₄), C₄⋊C₄.₂₁₆D₁₄, C₂₈⋊₂Q₈⋊₃₅C₂, D₁₄.₆₂(C₂×D₄), D₁₄.₁₆(C₂×Q₈), (D₇×C₄²).₉C₂, (C₂×Q₈).₁₄₂D₁₄, Dic₇.₁₀(C₂×D₄), C₁₄.₉₇(C₂²×D₄), Dic₇⋊Q₈⋊₂₅C₂, C₁₄.₄₆(C₂²×Q₈), (C₂×C₂₈).₁₀₀C₂³, (C₄×C₂₈).₂₀₈C₂², (C₂×C₁₄).₂₆₇C₂⁴, Dic₇⋊C₄.₅₉C₂², C₄⋊Dic₇.₂₅₀C₂², (Q₈×C₁₄).₁₃₄C₂², C₂².₂₈₈(C₂³×D₇), (C₄×Dic₇).₂₅₇C₂², (C₂×Dic₇).₁₃₉C₂³, (C₂²×D₇).₂₆₀C₂³, (C₂×Dic₁₄).₁₈₇C₂², C₇⋊₃(C₂×C₄⋊Q₈), (C₇×C₄⋊Q₈)⋊₉C₂, C₂.₇₀(C₂×D₄×D₇), (C₂×Q₈×D₇).₇C₂, C₂.₂₉(C₂×Q₈×D₇), (D₇×C₄⋊C₄).₁₂C₂, (C₂×C₄×D₇).₁₄₁C₂², (C₂×C₄).₉₂(C₂²×D₇), (C₇×C₄⋊C₄).₂₁₀C₂², SmallGroup(448,1176)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₇×C₄⋊Q₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂×C₄×D₇ — D₇×C₄² — D₇×C₄⋊Q₈

Lower central

C₇ — C₂×C₁₄ — D₇×C₄⋊Q₈

Upper central

C₁ — C₂² — C₄⋊Q₈

Generators and relations for D₇×C₄⋊Q₈
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=c^-1, ede^-1=d^-1 >

Subgroups: 1228 in 290 conjugacy classes, 131 normal (19 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, D₇, C₁₄, C₁₄, C₄², C₄², C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₄², C₂×C₄⋊C₄, C₄⋊Q₈, C₄⋊Q₈, C₂²×Q₈, Dic₁₄, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂×C₄⋊Q₈, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄×C₂₈, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×C₄×D₇, Q₈×D₇, Q₈×C₁₄, C₂₈⋊₂Q₈, D₇×C₄², C₂₈⋊Q₈, D₇×C₄⋊C₄, Dic₇⋊Q₈, C₇×C₄⋊Q₈, C₂×Q₈×D₇, D₇×C₄⋊Q₈
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂×D₄, C₂×Q₈, C₂⁴, D₁₄, C₄⋊Q₈, C₂²×D₄, C₂²×Q₈, C₂²×D₇, C₂×C₄⋊Q₈, D₄×D₇, Q₈×D₇, C₂³×D₇, C₂×D₄×D₇, C₂×Q₈×D₇, D₇×C₄⋊Q₈

Smallest permutation representation of D₇×C₄⋊Q₈
►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)

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

(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)

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

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

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

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

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

Matrix representation of D₇×C₄⋊Q₈ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	28	3	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	0	1	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	28	0	0	0
0	0	0	28	0	0
0	0	0	0	8	23
0	0	0	0	6	21

15	27	0	0	0	0
26	14	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	28	0	0	0	0
2	28	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	23	8
0	0	0	0	21	6

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,8,6,0,0,0,0,23,21],[15,26,0,0,0,0,27,14,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,2,0,0,0,0,28,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,23,21,0,0,0,0,8,6] >;

D₇×C₄⋊Q₈ in GAP, Magma, Sage, TeX

D_7\times C_4\rtimes Q_8

% in TeX

G:=Group("D7xC4:Q8");

// GroupNames label

G:=SmallGroup(448,1176);

// by ID

G=gap.SmallGroup(448,1176);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,232,100,570,185,80,18822]);

// 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^-1,e*d*e^-1=d^-1>;

// generators/relations

G = D7×C4⋊Q8 order 448 = 26·7

Direct product of D7 and C4⋊Q8

G = D₇×C₄⋊Q₈ order 448 = 2⁶·7

Direct product of D₇ and C₄⋊Q₈