C2^2xQ8xD7 - GroupNames

G = C₂²×Q₈×D₇ order 448 = 2⁶·7

Direct product of C₂², Q₈ and D₇

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

Aliases: C₂²×Q₈×D₇, C₁₄.₈C₂⁵, C₂₈.₄₃C₂⁴, D₁₄.₁₂C₂⁴, Dic₇.₄C₂⁴, Dic₁₄⋊₁₀C₂³, C₇⋊₂(Q₈×C₂³), (C₇×Q₈)⋊₆C₂³, C₁₄⋊₂(C₂²×Q₈), C₂.₉(D₇×C₂⁴), C₄.₄₃(C₂³×D₇), (Q₈×C₁₄)⋊₄₂C₂², (C₄×D₇).₃₂C₂³, (C₂×C₁₄).₃₂₈C₂⁴, (C₂×C₂₈).₅₆₄C₂³, (C₂²×C₄).₃₉₀D₁₄, C₂².₅₄(C₂³×D₇), (C₂×Dic₁₄)⋊₇₃C₂², (C₂²×Dic₁₄)⋊₂₄C₂, C₂³.₃₄₉(C₂²×D₇), (C₂²×C₂₈).₃₀₀C₂², (C₂²×C₁₄).₄₃₅C₂³, (C₂×Dic₇).₂₉₇C₂³, (C₂³×D₇).₁₂₄C₂², (C₂²×D₇).₂₆₂C₂³, (C₂²×Dic₇).₂₄₀C₂², (Q₈×C₂×C₁₄)⋊₉C₂, (C₂×C₁₄)⋊₁₀(C₂×Q₈), (D₇×C₂²×C₄).₁₀C₂, (C₂×C₄×D₇).₂₆₄C₂², (C₂×C₄).₆₄₄(C₂²×D₇), SmallGroup(448,1372)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂²×Q₈×D₇

Chief series

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

Lower central

C₇ — 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, bd=db, be=eb, bf=fb, dcd^-1=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 2932 in 850 conjugacy classes, 503 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, Q₈, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂³×C₄, C₂²×Q₈, C₂²×Q₈, Dic₁₄, C₄×D₇, C₂×Dic₇, C₂×C₂₈, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, Q₈×C₂³, C₂×Dic₁₄, C₂×C₄×D₇, Q₈×D₇, C₂²×Dic₇, C₂²×C₂₈, Q₈×C₁₄, C₂³×D₇, C₂²×Dic₁₄, D₇×C₂²×C₄, C₂×Q₈×D₇, Q₈×C₂×C₁₄, C₂²×Q₈×D₇
Quotients: C₁, C₂, C₂², Q₈, C₂³, D₇, C₂×Q₈, C₂⁴, D₁₄, C₂²×Q₈, C₂⁵, C₂²×D₇, Q₈×C₂³, Q₈×D₇, C₂³×D₇, C₂×Q₈×D₇, D₇×C₂⁴, C₂²×Q₈×D₇

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

Generators in S₂₂₄

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

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

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

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

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

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

100 conjugacy classes

class	1	2A	···	2G	2H	···	2O	4A	···	4L	4M	···	4X	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	7	···	7	2	···	2	14	···	14	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	Q₈	D₇	D₁₄	D₁₄	Q₈×D₇
kernel	C₂²×Q₈×D₇	C₂²×Dic₁₄	D₇×C₂²×C₄	C₂×Q₈×D₇	Q₈×C₂×C₁₄	C₂²×D₇	C₂²×Q₈	C₂²×C₄	C₂×Q₈	C₂²
# reps	1	3	3	24	1	8	3	9	36	12

Matrix representation of C₂²×Q₈×D₇ ►in GL₅(𝔽₂₉)

28	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	28	0
0	0	0	0	28

28	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	28	0
0	0	0	0	28

28	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	8	19
0	0	0	21	21

1	0	0	0	0
0	28	0	0	0
0	0	28	0	0
0	0	0	4	5
0	0	0	14	25

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

1	0	0	0	0
0	0	1	0	0
0	1	0	0	0
0	0	0	1	0
0	0	0	0	1

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,8,21,0,0,0,19,21],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,4,14,0,0,0,5,25],[1,0,0,0,0,0,0,28,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1] >;

C₂²×Q₈×D₇ in GAP, Magma, Sage, TeX

C_2^2\times Q_8\times D_7

% in TeX

G:=Group("C2^2xQ8xD7");

// GroupNames label

G:=SmallGroup(448,1372);

// by ID

G=gap.SmallGroup(448,1372);

# by ID

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

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

// generators/relations

G = C22×Q8×D7 order 448 = 26·7

Direct product of C22, Q8 and D7

G = C₂²×Q₈×D₇ order 448 = 2⁶·7

Direct product of C₂², Q₈ and D₇