C2^2xQ8:2D7 - GroupNames

G = C₂²×Q₈⋊₂D₇ order 448 = 2⁶·7

Direct product of C₂² and Q₈⋊₂D₇

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

Aliases: C₂²×Q₈⋊₂D₇, C₁₄.₉C₂⁵, D₂₈⋊₁₀C₂³, C₂₈.₄₄C₂⁴, D₁₄.₄C₂⁴, Dic₇.₁₅C₂⁴, (C₂×Q₈)⋊₃₇D₁₄, (C₄×D₇)⋊₆C₂³, Q₈⋊₆(C₂²×D₇), (C₇×Q₈)⋊₇C₂³, (C₂²×Q₈)⋊₁₁D₇, C₂.₁₀(D₇×C₂⁴), C₄.₄₄(C₂³×D₇), (C₂²×D₂₈)⋊₂₃C₂, (C₂×D₂₈)⋊₆₂C₂², (Q₈×C₁₄)⋊₄₃C₂², (C₂×C₁₄).₃₂₉C₂⁴, (C₂×C₂₈).₅₆₅C₂³, (C₂²×C₄).₃₉₁D₁₄, C₂².₅₅(C₂³×D₇), C₂³.₃₅₀(C₂²×D₇), (C₂²×C₂₈).₃₀₁C₂², (C₂²×C₁₄).₄₃₆C₂³, (C₂×Dic₇).₃₁₅C₂³, (C₂²×D₇).₂₄₇C₂³, (C₂³×D₇).₁₁₈C₂², (C₂²×Dic₇).₂₄₆C₂², C₁₄⋊₃(C₂×C₄○D₄), (Q₈×C₂×C₁₄)⋊₁₀C₂, C₇⋊₃(C₂²×C₄○D₄), (D₇×C₂²×C₄)⋊₁₀C₂, (C₂×C₄×D₇)⋊₆₀C₂², (C₂×C₁₄)⋊₁₉(C₄○D₄), (C₂×C₄).₆₄₅(C₂²×D₇), SmallGroup(448,1373)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂²×Q₈⋊₂D₇

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂³×D₇ — D₇×C₂²×C₄ — C₂²×Q₈⋊₂D₇

Lower central

C₇ — C₁₄ — C₂²×Q₈⋊₂D₇

Upper central

C₁ — C₂³ — C₂²×Q₈

Generators and relations for C₂²×Q₈⋊₂D₇
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, bd=db, be=eb, bf=fb, dcd^-1=fcf=c^-1, ce=ec, de=ed, df=fd, fef=e^-1 >

Subgroups: 3572 in 890 conjugacy classes, 463 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₄○D₄, C₂×C₄×D₇, C₂×D₂₈, Q₈⋊₂D₇, C₂²×Dic₇, C₂²×C₂₈, Q₈×C₁₄, C₂³×D₇, D₇×C₂²×C₄, C₂²×D₂₈, C₂×Q₈⋊₂D₇, Q₈×C₂×C₁₄, C₂²×Q₈⋊₂D₇
Quotients: C₁, C₂, C₂², C₂³, D₇, C₄○D₄, C₂⁴, D₁₄, C₂×C₄○D₄, C₂⁵, C₂²×D₇, C₂²×C₄○D₄, Q₈⋊₂D₇, C₂³×D₇, C₂×Q₈⋊₂D₇, D₇×C₂⁴, C₂²×Q₈⋊₂D₇

Smallest permutation representation of C₂²×Q₈⋊₂D₇
►On 224 points

Generators in S₂₂₄

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

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

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

(1 148 8 141)(2 149 9 142)(3 150 10 143)(4 151 11 144)(5 152 12 145)(6 153 13 146)(7 154 14 147)(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)(29 113 36 120)(30 114 37 121)(31 115 38 122)(32 116 39 123)(33 117 40 124)(34 118 41 125)(35 119 42 126)(43 127 50 134)(44 128 51 135)(45 129 52 136)(46 130 53 137)(47 131 54 138)(48 132 55 139)(49 133 56 140)(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)(85 169 92 176)(86 170 93 177)(87 171 94 178)(88 172 95 179)(89 173 96 180)(90 174 97 181)(91 175 98 182)(99 183 106 190)(100 184 107 191)(101 185 108 192)(102 186 109 193)(103 187 110 194)(104 188 111 195)(105 189 112 196)

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

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

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

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

100 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4L	4M	···	4T	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	14	···	14	2	···	2	7	···	7	2	2	2	2	···	2	4	···	4

100 irreducible representations

dim	1	1	1	1	1	2	2	2	2	4
type	+	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₇	C₄○D₄	D₁₄	D₁₄	Q₈⋊₂D₇
kernel	C₂²×Q₈⋊₂D₇	D₇×C₂²×C₄	C₂²×D₂₈	C₂×Q₈⋊₂D₇	Q₈×C₂×C₁₄	C₂²×Q₈	C₂×C₁₄	C₂²×C₄	C₂×Q₈	C₂²
# reps	1	3	3	24	1	3	8	9	36	12

Matrix representation of C₂²×Q₈⋊₂D₇ ►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

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	28	0
0	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	28	0	0
0	0	0	0	0	1
0	0	0	0	28	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	17	0
0	0	0	0	0	12

10	28	0	0	0	0
2	26	0	0	0	0
0	0	3	28	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	10	0	0	0	0
21	20	0	0	0	0
0	0	26	1	0	0
0	0	21	3	0	0
0	0	0	0	1	0
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,1,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,17,0,0,0,0,0,0,12],[10,2,0,0,0,0,28,26,0,0,0,0,0,0,3,1,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,21,0,0,0,0,10,20,0,0,0,0,0,0,26,21,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,28] >;

C₂²×Q₈⋊₂D₇ in GAP, Magma, Sage, TeX

C_2^2\times Q_8\rtimes_2D_7

% in TeX

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

// GroupNames label

G:=SmallGroup(448,1373);

// by ID

G=gap.SmallGroup(448,1373);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,136,1684,235,102,18822]);

// 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,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d^-1=f*c*f=c^-1,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = C22×Q8⋊2D7 order 448 = 26·7

Direct product of C22 and Q8⋊2D7

G = C₂²×Q₈⋊₂D₇ order 448 = 2⁶·7

Direct product of C₂² and Q₈⋊₂D₇