(C2^2xQ8):D7 - GroupNames

G = (C₂²×Q₈)⋊D₇ order 448 = 2⁶·7

2^nd semidirect product of C₂²×Q₈ and D₇ acting via D₇/C₇=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂²×Q₈)⋊₂D₇, (C₂²×D₇)⋊₂Q₈, (C₂×C₂₈).₃₀₆D₄, C₇⋊₃(C₂³⋊Q₈), C₁₄.₇₇C₂²≀C₂, C₂².₅₃(Q₈×D₇), (C₂²×C₄).₁₆₀D₁₄, C₁₄.₈₁(C₂²⋊Q₈), C₁₄.C₄²⋊₄₇C₂, C₂.₁₇(D₁₄⋊₃Q₈), C₁₄.₆₂(C₄.₄D₄), C₂.₁₁(C₂⁴⋊D₇), (C₂³×D₇).₂₆C₂², C₂³.₃₈₇(C₂²×D₇), C₂.₁₄(C₂₈.₂₃D₄), (C₂²×C₁₄).₃₇₁C₂³, (C₂²×C₂₈).₃₉₉C₂², C₂².₅₆(Q₈⋊₂D₇), (C₂²×Dic₇).₇₁C₂², (Q₈×C₂×C₁₄)⋊₁₂C₂, (C₂×C₁₄).₈₉(C₂×Q₈), (C₂×D₁₄⋊C₄).₂₅C₂, (C₂×C₁₄).₅₆₄(C₂×D₄), (C₂×C₄).₈₉(C₇⋊D₄), C₂².₂₂₃(C₂×C₇⋊D₄), (C₂×C₁₄).₁₉₅(C₄○D₄), SmallGroup(448,765)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — (C₂²×Q₈)⋊D₇

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂³×D₇ — C₂×D₁₄⋊C₄ — (C₂²×Q₈)⋊D₇

Lower central

C₇ — C₂²×C₁₄ — (C₂²×Q₈)⋊D₇

Upper central

C₁ — C₂³ — C₂²×Q₈

Generators and relations for (C₂²×Q₈)⋊D₇
G = < a,b,c,d,e,f | a²=b²=c⁴=e⁷=f²=1, d²=c², ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fdf=bd=db, be=eb, bf=fb, dcd^-1=c^-1, ce=ec, fcf=abc^-1, de=ed, fef=e^-1 >

Subgroups: 980 in 202 conjugacy classes, 65 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂⁴, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂.C₄², C₂×C₂²⋊C₄, C₂²×Q₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂³⋊Q₈, D₁₄⋊C₄, C₂²×Dic₇, C₂²×C₂₈, Q₈×C₁₄, C₂³×D₇, C₁₄.C₄², C₂×D₁₄⋊C₄, Q₈×C₂×C₁₄, (C₂²×Q₈)⋊D₇
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂×D₄, C₂×Q₈, C₄○D₄, D₁₄, C₂²≀C₂, C₂²⋊Q₈, C₄.₄D₄, C₇⋊D₄, C₂²×D₇, C₂³⋊Q₈, Q₈×D₇, Q₈⋊₂D₇, C₂×C₇⋊D₄, D₁₄⋊₃Q₈, C₂₈.₂₃D₄, C₂⁴⋊D₇, (C₂²×Q₈)⋊D₇

Smallest permutation representation of (C₂²×Q₈)⋊D₇
►On 224 points

Generators in S₂₂₄

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

(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(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 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)(113 127)(114 128)(115 129)(116 130)(117 131)(118 132)(119 133)(120 134)(121 135)(122 136)(123 137)(124 138)(125 139)(126 140)(141 155)(142 156)(143 157)(144 158)(145 159)(146 160)(147 161)(148 162)(149 163)(150 164)(151 165)(152 166)(153 167)(154 168)(169 183)(170 184)(171 185)(172 186)(173 187)(174 188)(175 189)(176 190)(177 191)(178 192)(179 193)(180 194)(181 195)(182 196)(197 211)(198 212)(199 213)(200 214)(201 215)(202 216)(203 217)(204 218)(205 219)(206 220)(207 221)(208 222)(209 223)(210 224)

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

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

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

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

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

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

82 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4F	4G	···	4L	7A	7B	7C	14A	···	14U	28A	···	28AJ
order	1	2	···	2	2	2	4	···	4	4	···	4	7	7	7	14	···	14	28	···	28
size	1	1	···	1	28	28	4	···	4	28	···	28	2	2	2	2	···	2	4	···	4

82 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	4	4
type	+	+	+	+	+	-	+		+		-	+
image	C₁	C₂	C₂	C₂	D₄	Q₈	D₇	C₄○D₄	D₁₄	C₇⋊D₄	Q₈×D₇	Q₈⋊₂D₇
kernel	(C₂²×Q₈)⋊D₇	C₁₄.C₄²	C₂×D₁₄⋊C₄	Q₈×C₂×C₁₄	C₂×C₂₈	C₂²×D₇	C₂²×Q₈	C₂×C₁₄	C₂²×C₄	C₂×C₄	C₂²	C₂²
# reps	1	3	3	1	6	2	3	6	9	36	3	9

Matrix representation of (C₂²×Q₈)⋊D₇ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	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

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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	18	14	0	0
0	0	8	11	0	0
0	0	0	0	23	5
0	0	0	0	10	6

11	2	0	0	0	0
27	18	0	0	0	0
0	0	18	14	0	0
0	0	8	11	0	0
0	0	0	0	17	0
0	0	0	0	6	12

0	1	0	0	0	0
28	18	0	0	0	0
0	0	11	1	0	0
0	0	13	25	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	1	0	0	0	0
1	0	0	0	0	0
0	0	25	28	0	0
0	0	15	4	0	0
0	0	0	0	28	0
0	0	0	0	15	1

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,8,0,0,0,0,14,11,0,0,0,0,0,0,23,10,0,0,0,0,5,6],[11,27,0,0,0,0,2,18,0,0,0,0,0,0,18,8,0,0,0,0,14,11,0,0,0,0,0,0,17,6,0,0,0,0,0,12],[0,28,0,0,0,0,1,18,0,0,0,0,0,0,11,13,0,0,0,0,1,25,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,25,15,0,0,0,0,28,4,0,0,0,0,0,0,28,15,0,0,0,0,0,1] >;

(C₂²×Q₈)⋊D₇ in GAP, Magma, Sage, TeX

(C_2^2\times Q_8)\rtimes D_7

% in TeX

G:=Group("(C2^2xQ8):D7");

// GroupNames label

G:=SmallGroup(448,765);

// by ID

G=gap.SmallGroup(448,765);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,232,254,387,184,18822]);

// Polycyclic

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

// generators/relations

G = (C22×Q8)⋊D7 order 448 = 26·7

2nd semidirect product of C22×Q8 and D7 acting via D7/C7=C2

G = (C₂²×Q₈)⋊D₇ order 448 = 2⁶·7

2^nd semidirect product of C₂²×Q₈ and D₇ acting via D₇/C₇=C₂