(Q8xDic7):C2 - GroupNames

G = (Q₈×Dic₇)⋊C₂ order 448 = 2⁶·7

10^th semidirect product of Q₈×Dic₇ and C₂ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈⋊Q₈⋊₂₂C₂, (C₂×Dic₇)⋊₉Q₈, C₄⋊C₄.₁₈₆D₁₄, (Q₈×Dic₇)⋊₁₀C₂, C₂².₅(Q₈×D₇), Dic₇.₃(C₂×Q₈), C₂²⋊Q₈.₁₀D₇, Dic₇.Q₈⋊₁₅C₂, (C₂×C₂₈).₄₈C₂³, (C₂×Q₈).₁₂₁D₁₄, C₂²⋊C₄.₅₂D₁₄, Dic₇⋊₃Q₈⋊₂₃C₂, Dic₇⋊Q₈⋊₁₁C₂, C₂₈.₂₀₇(C₄○D₄), C₄.₇₀(D₄⋊₂D₇), C₁₄.₃₂(C₂²×Q₈), (C₂×C₁₄).₁₆₆C₂⁴, Dic₇.₆(C₄○D₄), (C₂²×C₄).₃₆₉D₁₄, C₂₈.₄₈D₄.₁₅C₂, Dic₇⋊C₄.₂₂C₂², C₄⋊Dic₇.₂₁₁C₂², C₂²⋊Dic₁₄.₂C₂, (Q₈×C₁₄).₁₀₁C₂², (C₂×Dic₇).₈₃C₂³, C₂².₁₈₇(C₂³×D₇), C₂³.₁₈₄(C₂²×D₇), C₂³.D₇.₃₀C₂², (C₂²×C₂₈).₂₄₇C₂², (C₂²×C₁₄).₁₉₄C₂³, C₇⋊₄(C₂³.₃₇C₂³), (C₄×Dic₇).₂₅₅C₂², C₂³.₁₁D₁₄.₁C₂, (C₂×Dic₁₄).₁₅₇C₂², (C₂²×Dic₇).₂₂₂C₂², C₂.₁₅(C₂×Q₈×D₇), C₂.₄₅(D₇×C₄○D₄), (C₂×C₁₄).₅(C₂×Q₈), C₁₄.₈₈(C₂×C₄○D₄), (C₂×C₄×Dic₇).₁₅C₂, (C₇×C₂²⋊Q₈).₇C₂, C₂.₄₃(C₂×D₄⋊₂D₇), (C₇×C₄⋊C₄).₁₅₂C₂², (C₂×C₄).₂₉₅(C₂²×D₇), (C₇×C₂²⋊C₄).₂₁C₂², SmallGroup(448,1075)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — (Q₈×Dic₇)⋊C₂

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×Dic₇ — C₂²×Dic₇ — C₂×C₄×Dic₇ — (Q₈×Dic₇)⋊C₂

Lower central

C₇ — C₂×C₁₄ — (Q₈×Dic₇)⋊C₂

Upper central

C₁ — C₂² — C₂²⋊Q₈

Generators and relations for (Q₈×Dic₇)⋊C₂
G = < a,b,c,d,e | a⁴=c¹⁴=e²=1, b²=a², d²=c⁷, bab^-1=a^-1, ac=ca, ad=da, eae=ac⁷, bc=cb, bd=db, be=eb, dcd^-1=c^-1, ce=ec, de=ed >

Subgroups: 796 in 222 conjugacy classes, 109 normal (43 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₁₄, C₁₄, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, Dic₇, Dic₇, C₂₈, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₄², C₄²⋊C₂, C₄×Q₈, C₂²⋊Q₈, C₂²⋊Q₈, C₄².C₂, C₄⋊Q₈, Dic₁₄, C₂×Dic₇, C₂×Dic₇, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×C₁₄, C₂³.₃₇C₂³, C₄×Dic₇, C₄×Dic₇, Dic₇⋊C₄, C₄⋊Dic₇, C₄⋊Dic₇, C₂³.D₇, C₇×C₂²⋊C₄, C₇×C₄⋊C₄, C₇×C₄⋊C₄, C₂×Dic₁₄, C₂×Dic₁₄, C₂²×Dic₇, C₂²×C₂₈, Q₈×C₁₄, C₂³.₁₁D₁₄, C₂²⋊Dic₁₄, Dic₇⋊₃Q₈, Dic₇⋊₃Q₈, C₂₈⋊Q₈, Dic₇.Q₈, C₂×C₄×Dic₇, C₂₈.₄₈D₄, Dic₇⋊Q₈, Q₈×Dic₇, C₇×C₂²⋊Q₈, (Q₈×Dic₇)⋊C₂
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₄○D₄, C₂⁴, D₁₄, C₂²×Q₈, C₂×C₄○D₄, C₂²×D₇, C₂³.₃₇C₂³, D₄⋊₂D₇, Q₈×D₇, C₂³×D₇, C₂×D₄⋊₂D₇, C₂×Q₈×D₇, D₇×C₄○D₄, (Q₈×Dic₇)⋊C₂

Smallest permutation representation of (Q₈×Dic₇)⋊C₂
►On 224 points

Generators in S₂₂₄

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

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

(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)

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

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

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

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

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

70 conjugacy classes

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

70 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	-	+			+	+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	Q₈	D₇	C₄○D₄	C₄○D₄	D₁₄	D₁₄	D₁₄	D₁₄	D₄⋊₂D₇	Q₈×D₇	D₇×C₄○D₄
kernel	(Q₈×Dic₇)⋊C₂	C₂³.₁₁D₁₄	C₂²⋊Dic₁₄	Dic₇⋊₃Q₈	C₂₈⋊Q₈	Dic₇.Q₈	C₂×C₄×Dic₇	C₂₈.₄₈D₄	Dic₇⋊Q₈	Q₈×Dic₇	C₇×C₂²⋊Q₈	C₂×Dic₇	C₂²⋊Q₈	Dic₇	C₂₈	C₂²⋊C₄	C₄⋊C₄	C₂²×C₄	C₂×Q₈	C₄	C₂²	C₂
# reps	1	2	2	3	1	2	1	1	1	1	1	4	3	4	4	6	9	3	3	6	6	6

Matrix representation of (Q₈×Dic₇)⋊C₂ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	28	24	0	0
0	0	0	1	0	0
0	0	0	0	0	1
0	0	0	0	28	0

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	17	0
0	0	0	0	0	12

7	28	0	0	0	0
1	0	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

19	22	0	0	0	0
10	10	0	0	0	0
0	0	17	0	0	0
0	0	0	17	0	0
0	0	0	0	17	0
0	0	0	0	0	17

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	17	28	0	0
0	0	0	0	28	0
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,24,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,12],[7,1,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[19,10,0,0,0,0,22,10,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1] >;

(Q₈×Dic₇)⋊C₂ in GAP, Magma, Sage, TeX

(Q_8\times {\rm Dic}_7)\rtimes C_2

% in TeX

G:=Group("(Q8xDic7):C2");

// GroupNames label

G:=SmallGroup(448,1075);

// by ID

G=gap.SmallGroup(448,1075);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,100,570,185,80,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^14=e^2=1,b^2=a^2,d^2=c^7,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,e*a*e=a*c^7,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,d*e=e*d>;

// generators/relations

G = (Q8×Dic7)⋊C2 order 448 = 26·7

10th semidirect product of Q8×Dic7 and C2 acting faithfully

G = (Q₈×Dic₇)⋊C₂ order 448 = 2⁶·7

10^th semidirect product of Q₈×Dic₇ and C₂ acting faithfully