C2xSD16:D7 - GroupNames

G = C₂×SD₁₆⋊D₇ order 448 = 2⁶·7

Direct product of C₂ and SD₁₆⋊D₇

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

Aliases: C₂×SD₁₆⋊D₇, SD₁₆⋊₉D₁₄, C₂₈.₇C₂⁴, C₅₆.₃₅C₂³, Dic₂₈⋊₁₇C₂², Dic₁₄.₃C₂³, C₄.₄₄(D₄×D₇), C₇⋊C₈.₂C₂³, (C₂×SD₁₆)⋊₅D₇, (C₄×D₇).₁₆D₄, C₂₈.₈₂(C₂×D₄), (Q₈×D₇)⋊₆C₂², C₄.₇(C₂³×D₇), (C₁₄×SD₁₆)⋊₆C₂, D₁₄.₅₁(C₂×D₄), (C₂×C₈).₁₀₃D₁₄, C₈⋊D₇⋊₉C₂², (C₇×D₄).₅C₂³, C₇⋊Q₁₆⋊₆C₂², D₄.₅(C₂²×D₇), (C₄×D₇).₄C₂³, C₈.₁₁(C₂²×D₇), (C₂×Dic₂₈)⋊₂₆C₂, (C₂×D₄).₁₈₃D₁₄, Q₈.₁(C₂²×D₇), (C₇×Q₈).₁C₂³, C₁₄⋊₂(C₈.C₂²), D₄.D₇⋊₁₀C₂², (C₂×Q₈).₁₅₀D₁₄, Dic₇.₅₆(C₂×D₄), (C₇×SD₁₆)⋊₉C₂², (C₂²×D₇).₉₉D₄, C₂².₁₄₀(D₄×D₇), (C₂×C₂₈).₅₂₄C₂³, (C₂×C₅₆).₁₁₇C₂², (C₂×Dic₇).₁₉₃D₄, D₄⋊₂D₇.₄C₂², C₁₄.₁₀₈(C₂²×D₄), (D₄×C₁₄).₁₆₅C₂², (Q₈×C₁₄).₁₄₇C₂², (C₂×Dic₁₄).₁₉₆C₂², (C₂×Q₈×D₇)⋊₁₅C₂, C₂.₈₁(C₂×D₄×D₇), C₇⋊₂(C₂×C₈.C₂²), (C₂×C₈⋊D₇)⋊₅C₂, (C₂×D₄.D₇)⋊₂₈C₂, (C₂×C₇⋊Q₁₆)⋊₂₅C₂, (C₂×C₁₄).₃₉₇(C₂×D₄), (C₂×C₇⋊C₈).₁₈₀C₂², (C₂×C₄×D₇).₁₅₇C₂², (C₂×D₄⋊₂D₇).₁₀C₂, (C₂×C₄).₆₁₃(C₂²×D₇), SmallGroup(448,1213)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₈ — C₂×SD₁₆⋊D₇

Chief series

C₁ — C₇ — C₁₄ — C₂₈ — C₄×D₇ — C₂×C₄×D₇ — C₂×Q₈×D₇ — C₂×SD₁₆⋊D₇

Lower central

C₇ — C₁₄ — C₂₈ — C₂×SD₁₆⋊D₇

Upper central

C₁ — C₂² — C₂×C₄ — C₂×SD₁₆

Generators and relations for C₂×SD₁₆⋊D₇
G = < a,b,c,d,e | a²=b⁸=c²=d⁷=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b³, bd=db, ebe=b⁵, cd=dc, ece=b⁴c, ede=d^-1 >

Subgroups: 1156 in 258 conjugacy classes, 103 normal (33 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, SD₁₆, Q₁₆, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, Dic₇, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×M₄(2), C₂×SD₁₆, C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂²×Q₈, C₂×C₄○D₄, C₇⋊C₈, C₅₆, Dic₁₄, Dic₁₄, C₄×D₇, C₄×D₇, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×Q₈, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₂×C₈.C₂², C₈⋊D₇, Dic₂₈, C₂×C₇⋊C₈, D₄.D₇, C₇⋊Q₁₆, C₂×C₅₆, C₇×SD₁₆, C₂×Dic₁₄, C₂×Dic₁₄, C₂×C₄×D₇, C₂×C₄×D₇, D₄⋊₂D₇, D₄⋊₂D₇, Q₈×D₇, Q₈×D₇, C₂²×Dic₇, C₂×C₇⋊D₄, D₄×C₁₄, Q₈×C₁₄, C₂×C₈⋊D₇, C₂×Dic₂₈, SD₁₆⋊D₇, C₂×D₄.D₇, C₂×C₇⋊Q₁₆, C₁₄×SD₁₆, C₂×D₄⋊₂D₇, C₂×Q₈×D₇, C₂×SD₁₆⋊D₇
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₈.C₂², C₂²×D₄, C₂²×D₇, C₂×C₈.C₂², D₄×D₇, C₂³×D₇, SD₁₆⋊D₇, C₂×D₄×D₇, C₂×SD₁₆⋊D₇

Smallest permutation representation of C₂×SD₁₆⋊D₇
►On 224 points

Generators in S₂₂₄

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

(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)

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

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

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

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	7A	7B	7C	8A	8B	8C	8D	14A	···	14I	14J	···	14O	28A	···	28F	28G	···	28L	56A	···	56L
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	4	7	7	7	8	8	8	8	14	···	14	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	1	1	4	4	14	14	2	2	4	4	14	14	28	28	28	28	2	2	2	4	4	28	28	2	···	2	8	···	8	4	···	4	8	···	8	4	···	4

64 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	D₄	D₄	D₇	D₁₄	D₁₄	D₁₄	D₁₄	C₈.C₂²	D₄×D₇	D₄×D₇	SD₁₆⋊D₇
kernel	C₂×SD₁₆⋊D₇	C₂×C₈⋊D₇	C₂×Dic₂₈	SD₁₆⋊D₇	C₂×D₄.D₇	C₂×C₇⋊Q₁₆	C₁₄×SD₁₆	C₂×D₄⋊₂D₇	C₂×Q₈×D₇	C₄×D₇	C₂×Dic₇	C₂²×D₇	C₂×SD₁₆	C₂×C₈	SD₁₆	C₂×D₄	C₂×Q₈	C₁₄	C₄	C₂²	C₂
# reps	1	1	1	8	1	1	1	1	1	2	1	1	3	3	12	3	3	2	3	3	12

Matrix representation of C₂×SD₁₆⋊D₇ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	112	0
0	0	0	0	0	112

60	18	0	0	0	0
95	53	0	0	0	0
0	0	0	0	37	27
0	0	0	0	12	76
0	0	38	43	37	27
0	0	107	75	12	76

0	1	0	0	0	0
1	0	0	0	0	0
0	0	86	20	34	92
0	0	34	27	66	79
0	0	103	66	27	93
0	0	67	10	79	86

1	0	0	0	0	0
0	1	0	0	0	0
0	0	103	112	0	0
0	0	100	89	0	0
0	0	0	0	103	112
0	0	0	0	100	89

1	0	0	0	0	0
0	1	0	0	0	0
0	0	103	25	0	0
0	0	100	10	0	0
0	0	0	0	103	25
0	0	0	0	100	10

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[60,95,0,0,0,0,18,53,0,0,0,0,0,0,0,0,38,107,0,0,0,0,43,75,0,0,37,12,37,12,0,0,27,76,27,76],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,86,34,103,67,0,0,20,27,66,10,0,0,34,66,27,79,0,0,92,79,93,86],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,100,0,0,0,0,112,89,0,0,0,0,0,0,103,100,0,0,0,0,112,89],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,100,0,0,0,0,25,10,0,0,0,0,0,0,103,100,0,0,0,0,25,10] >;

C₂×SD₁₆⋊D₇ in GAP, Magma, Sage, TeX

C_2\times {\rm SD}_{16}\rtimes D_7

% in TeX

G:=Group("C2xSD16:D7");

// GroupNames label

G:=SmallGroup(448,1213);

// by ID

G=gap.SmallGroup(448,1213);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,184,1123,185,438,235,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^7=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^3,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;

// generators/relations

G = C2×SD16⋊D7 order 448 = 26·7

Direct product of C2 and SD16⋊D7

G = C₂×SD₁₆⋊D₇ order 448 = 2⁶·7

Direct product of C₂ and SD₁₆⋊D₇