C2xC5^3:C2 - GroupNames

G = C₂×C₅³⋊C₂ order 500 = 2²·5³

Direct product of C₂ and C₅³⋊C₂

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₂×C₅³⋊C₂, C₅²⋊₈D₁₀, C₅³⋊₆C₂², C₁₀⋊(C₅⋊D₅), (C₅×C₁₀)⋊₄D₅, (C₅²×C₁₀)⋊₃C₂, C₅⋊₂(C₂×C₅⋊D₅), SmallGroup(500,55)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅³ — C₂×C₅³⋊C₂

Chief series

C₁ — C₅ — C₅² — C₅³ — C₅³⋊C₂ — C₂×C₅³⋊C₂

Lower central

C₅³ — C₂×C₅³⋊C₂

Upper central

C₁ — C₂

Generators and relations for C₂×C₅³⋊C₂
G = < a,b,c,d,e | a²=b⁵=c⁵=d⁵=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=b^-1, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 3296 in 320 conjugacy classes, 131 normal (5 characteristic)
C₁, C₂, C₂, C₂², C₅, D₅, C₁₀, D₁₀, C₅², C₅⋊D₅, C₅×C₁₀, C₂×C₅⋊D₅, C₅³, C₅³⋊C₂, C₅²×C₁₀, C₂×C₅³⋊C₂
Quotients: C₁, C₂, C₂², D₅, D₁₀, C₅⋊D₅, C₂×C₅⋊D₅, C₅³⋊C₂, C₂×C₅³⋊C₂

Smallest permutation representation of C₂×C₅³⋊C₂
►On 250 points

Generators in S₂₅₀

(1 134)(2 135)(3 131)(4 132)(5 133)(6 227)(7 228)(8 229)(9 230)(10 226)(11 136)(12 137)(13 138)(14 139)(15 140)(16 141)(17 142)(18 143)(19 144)(20 145)(21 146)(22 147)(23 148)(24 149)(25 150)(26 130)(27 126)(28 127)(29 128)(30 129)(31 156)(32 157)(33 158)(34 159)(35 160)(36 161)(37 162)(38 163)(39 164)(40 165)(41 166)(42 167)(43 168)(44 169)(45 170)(46 171)(47 172)(48 173)(49 174)(50 175)(51 155)(52 151)(53 152)(54 153)(55 154)(56 181)(57 182)(58 183)(59 184)(60 185)(61 186)(62 187)(63 188)(64 189)(65 190)(66 191)(67 192)(68 193)(69 194)(70 195)(71 196)(72 197)(73 198)(74 199)(75 200)(76 180)(77 176)(78 177)(79 178)(80 179)(81 206)(82 207)(83 208)(84 209)(85 210)(86 211)(87 212)(88 213)(89 214)(90 215)(91 216)(92 217)(93 218)(94 219)(95 220)(96 221)(97 222)(98 223)(99 224)(100 225)(101 205)(102 201)(103 202)(104 203)(105 204)(106 231)(107 232)(108 233)(109 234)(110 235)(111 236)(112 237)(113 238)(114 239)(115 240)(116 241)(117 242)(118 243)(119 244)(120 245)(121 246)(122 247)(123 248)(124 249)(125 250)

(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)(221 222 223 224 225)(226 227 228 229 230)(231 232 233 234 235)(236 237 238 239 240)(241 242 243 244 245)(246 247 248 249 250)

(1 7 84 59 34)(2 8 85 60 35)(3 9 81 56 31)(4 10 82 57 32)(5 6 83 58 33)(11 110 86 61 36)(12 106 87 62 37)(13 107 88 63 38)(14 108 89 64 39)(15 109 90 65 40)(16 115 91 66 41)(17 111 92 67 42)(18 112 93 68 43)(19 113 94 69 44)(20 114 95 70 45)(21 120 96 71 46)(22 116 97 72 47)(23 117 98 73 48)(24 118 99 74 49)(25 119 100 75 50)(26 250 101 76 51)(27 246 102 77 52)(28 247 103 78 53)(29 248 104 79 54)(30 249 105 80 55)(121 201 176 151 126)(122 202 177 152 127)(123 203 178 153 128)(124 204 179 154 129)(125 205 180 155 130)(131 230 206 181 156)(132 226 207 182 157)(133 227 208 183 158)(134 228 209 184 159)(135 229 210 185 160)(136 235 211 186 161)(137 231 212 187 162)(138 232 213 188 163)(139 233 214 189 164)(140 234 215 190 165)(141 240 216 191 166)(142 236 217 192 167)(143 237 218 193 168)(144 238 219 194 169)(145 239 220 195 170)(146 245 221 196 171)(147 241 222 197 172)(148 242 223 198 173)(149 243 224 199 174)(150 244 225 200 175)

(1 124 24 19 14)(2 125 25 20 15)(3 121 21 16 11)(4 122 22 17 12)(5 123 23 18 13)(6 203 117 112 107)(7 204 118 113 108)(8 205 119 114 109)(9 201 120 115 110)(10 202 116 111 106)(26 175 170 165 160)(27 171 166 161 156)(28 172 167 162 157)(29 173 168 163 158)(30 174 169 164 159)(31 126 46 41 36)(32 127 47 42 37)(33 128 48 43 38)(34 129 49 44 39)(35 130 50 45 40)(51 200 195 190 185)(52 196 191 186 181)(53 197 192 187 182)(54 198 193 188 183)(55 199 194 189 184)(56 151 71 66 61)(57 152 72 67 62)(58 153 73 68 63)(59 154 74 69 64)(60 155 75 70 65)(76 225 220 215 210)(77 221 216 211 206)(78 222 217 212 207)(79 223 218 213 208)(80 224 219 214 209)(81 176 96 91 86)(82 177 97 92 87)(83 178 98 93 88)(84 179 99 94 89)(85 180 100 95 90)(101 244 239 234 229)(102 245 240 235 230)(103 241 236 231 226)(104 242 237 232 227)(105 243 238 233 228)(131 246 146 141 136)(132 247 147 142 137)(133 248 148 143 138)(134 249 149 144 139)(135 250 150 145 140)

(2 5)(3 4)(6 35)(7 34)(8 33)(9 32)(10 31)(11 122)(12 121)(13 125)(14 124)(15 123)(16 22)(17 21)(18 25)(19 24)(20 23)(26 232)(27 231)(28 235)(29 234)(30 233)(36 202)(37 201)(38 205)(39 204)(40 203)(41 116)(42 120)(43 119)(44 118)(45 117)(46 111)(47 115)(48 114)(49 113)(50 112)(51 213)(52 212)(53 211)(54 215)(55 214)(56 82)(57 81)(58 85)(59 84)(60 83)(61 177)(62 176)(63 180)(64 179)(65 178)(66 97)(67 96)(68 100)(69 99)(70 98)(71 92)(72 91)(73 95)(74 94)(75 93)(76 188)(77 187)(78 186)(79 190)(80 189)(86 152)(87 151)(88 155)(89 154)(90 153)(101 163)(102 162)(103 161)(104 165)(105 164)(106 126)(107 130)(108 129)(109 128)(110 127)(131 132)(133 135)(136 247)(137 246)(138 250)(139 249)(140 248)(141 147)(142 146)(143 150)(144 149)(145 148)(156 226)(157 230)(158 229)(159 228)(160 227)(166 241)(167 245)(168 244)(169 243)(170 242)(171 236)(172 240)(173 239)(174 238)(175 237)(181 207)(182 206)(183 210)(184 209)(185 208)(191 222)(192 221)(193 225)(194 224)(195 223)(196 217)(197 216)(198 220)(199 219)(200 218)

G:=sub<Sym(250)| (1,134)(2,135)(3,131)(4,132)(5,133)(6,227)(7,228)(8,229)(9,230)(10,226)(11,136)(12,137)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,130)(27,126)(28,127)(29,128)(30,129)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,155)(52,151)(53,152)(54,153)(55,154)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,196)(72,197)(73,198)(74,199)(75,200)(76,180)(77,176)(78,177)(79,178)(80,179)(81,206)(82,207)(83,208)(84,209)(85,210)(86,211)(87,212)(88,213)(89,214)(90,215)(91,216)(92,217)(93,218)(94,219)(95,220)(96,221)(97,222)(98,223)(99,224)(100,225)(101,205)(102,201)(103,202)(104,203)(105,204)(106,231)(107,232)(108,233)(109,234)(110,235)(111,236)(112,237)(113,238)(114,239)(115,240)(116,241)(117,242)(118,243)(119,244)(120,245)(121,246)(122,247)(123,248)(124,249)(125,250), (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)(221,222,223,224,225)(226,227,228,229,230)(231,232,233,234,235)(236,237,238,239,240)(241,242,243,244,245)(246,247,248,249,250), (1,7,84,59,34)(2,8,85,60,35)(3,9,81,56,31)(4,10,82,57,32)(5,6,83,58,33)(11,110,86,61,36)(12,106,87,62,37)(13,107,88,63,38)(14,108,89,64,39)(15,109,90,65,40)(16,115,91,66,41)(17,111,92,67,42)(18,112,93,68,43)(19,113,94,69,44)(20,114,95,70,45)(21,120,96,71,46)(22,116,97,72,47)(23,117,98,73,48)(24,118,99,74,49)(25,119,100,75,50)(26,250,101,76,51)(27,246,102,77,52)(28,247,103,78,53)(29,248,104,79,54)(30,249,105,80,55)(121,201,176,151,126)(122,202,177,152,127)(123,203,178,153,128)(124,204,179,154,129)(125,205,180,155,130)(131,230,206,181,156)(132,226,207,182,157)(133,227,208,183,158)(134,228,209,184,159)(135,229,210,185,160)(136,235,211,186,161)(137,231,212,187,162)(138,232,213,188,163)(139,233,214,189,164)(140,234,215,190,165)(141,240,216,191,166)(142,236,217,192,167)(143,237,218,193,168)(144,238,219,194,169)(145,239,220,195,170)(146,245,221,196,171)(147,241,222,197,172)(148,242,223,198,173)(149,243,224,199,174)(150,244,225,200,175), (1,124,24,19,14)(2,125,25,20,15)(3,121,21,16,11)(4,122,22,17,12)(5,123,23,18,13)(6,203,117,112,107)(7,204,118,113,108)(8,205,119,114,109)(9,201,120,115,110)(10,202,116,111,106)(26,175,170,165,160)(27,171,166,161,156)(28,172,167,162,157)(29,173,168,163,158)(30,174,169,164,159)(31,126,46,41,36)(32,127,47,42,37)(33,128,48,43,38)(34,129,49,44,39)(35,130,50,45,40)(51,200,195,190,185)(52,196,191,186,181)(53,197,192,187,182)(54,198,193,188,183)(55,199,194,189,184)(56,151,71,66,61)(57,152,72,67,62)(58,153,73,68,63)(59,154,74,69,64)(60,155,75,70,65)(76,225,220,215,210)(77,221,216,211,206)(78,222,217,212,207)(79,223,218,213,208)(80,224,219,214,209)(81,176,96,91,86)(82,177,97,92,87)(83,178,98,93,88)(84,179,99,94,89)(85,180,100,95,90)(101,244,239,234,229)(102,245,240,235,230)(103,241,236,231,226)(104,242,237,232,227)(105,243,238,233,228)(131,246,146,141,136)(132,247,147,142,137)(133,248,148,143,138)(134,249,149,144,139)(135,250,150,145,140), (2,5)(3,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,122)(12,121)(13,125)(14,124)(15,123)(16,22)(17,21)(18,25)(19,24)(20,23)(26,232)(27,231)(28,235)(29,234)(30,233)(36,202)(37,201)(38,205)(39,204)(40,203)(41,116)(42,120)(43,119)(44,118)(45,117)(46,111)(47,115)(48,114)(49,113)(50,112)(51,213)(52,212)(53,211)(54,215)(55,214)(56,82)(57,81)(58,85)(59,84)(60,83)(61,177)(62,176)(63,180)(64,179)(65,178)(66,97)(67,96)(68,100)(69,99)(70,98)(71,92)(72,91)(73,95)(74,94)(75,93)(76,188)(77,187)(78,186)(79,190)(80,189)(86,152)(87,151)(88,155)(89,154)(90,153)(101,163)(102,162)(103,161)(104,165)(105,164)(106,126)(107,130)(108,129)(109,128)(110,127)(131,132)(133,135)(136,247)(137,246)(138,250)(139,249)(140,248)(141,147)(142,146)(143,150)(144,149)(145,148)(156,226)(157,230)(158,229)(159,228)(160,227)(166,241)(167,245)(168,244)(169,243)(170,242)(171,236)(172,240)(173,239)(174,238)(175,237)(181,207)(182,206)(183,210)(184,209)(185,208)(191,222)(192,221)(193,225)(194,224)(195,223)(196,217)(197,216)(198,220)(199,219)(200,218)>;

G:=Group( (1,134)(2,135)(3,131)(4,132)(5,133)(6,227)(7,228)(8,229)(9,230)(10,226)(11,136)(12,137)(13,138)(14,139)(15,140)(16,141)(17,142)(18,143)(19,144)(20,145)(21,146)(22,147)(23,148)(24,149)(25,150)(26,130)(27,126)(28,127)(29,128)(30,129)(31,156)(32,157)(33,158)(34,159)(35,160)(36,161)(37,162)(38,163)(39,164)(40,165)(41,166)(42,167)(43,168)(44,169)(45,170)(46,171)(47,172)(48,173)(49,174)(50,175)(51,155)(52,151)(53,152)(54,153)(55,154)(56,181)(57,182)(58,183)(59,184)(60,185)(61,186)(62,187)(63,188)(64,189)(65,190)(66,191)(67,192)(68,193)(69,194)(70,195)(71,196)(72,197)(73,198)(74,199)(75,200)(76,180)(77,176)(78,177)(79,178)(80,179)(81,206)(82,207)(83,208)(84,209)(85,210)(86,211)(87,212)(88,213)(89,214)(90,215)(91,216)(92,217)(93,218)(94,219)(95,220)(96,221)(97,222)(98,223)(99,224)(100,225)(101,205)(102,201)(103,202)(104,203)(105,204)(106,231)(107,232)(108,233)(109,234)(110,235)(111,236)(112,237)(113,238)(114,239)(115,240)(116,241)(117,242)(118,243)(119,244)(120,245)(121,246)(122,247)(123,248)(124,249)(125,250), (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)(221,222,223,224,225)(226,227,228,229,230)(231,232,233,234,235)(236,237,238,239,240)(241,242,243,244,245)(246,247,248,249,250), (1,7,84,59,34)(2,8,85,60,35)(3,9,81,56,31)(4,10,82,57,32)(5,6,83,58,33)(11,110,86,61,36)(12,106,87,62,37)(13,107,88,63,38)(14,108,89,64,39)(15,109,90,65,40)(16,115,91,66,41)(17,111,92,67,42)(18,112,93,68,43)(19,113,94,69,44)(20,114,95,70,45)(21,120,96,71,46)(22,116,97,72,47)(23,117,98,73,48)(24,118,99,74,49)(25,119,100,75,50)(26,250,101,76,51)(27,246,102,77,52)(28,247,103,78,53)(29,248,104,79,54)(30,249,105,80,55)(121,201,176,151,126)(122,202,177,152,127)(123,203,178,153,128)(124,204,179,154,129)(125,205,180,155,130)(131,230,206,181,156)(132,226,207,182,157)(133,227,208,183,158)(134,228,209,184,159)(135,229,210,185,160)(136,235,211,186,161)(137,231,212,187,162)(138,232,213,188,163)(139,233,214,189,164)(140,234,215,190,165)(141,240,216,191,166)(142,236,217,192,167)(143,237,218,193,168)(144,238,219,194,169)(145,239,220,195,170)(146,245,221,196,171)(147,241,222,197,172)(148,242,223,198,173)(149,243,224,199,174)(150,244,225,200,175), (1,124,24,19,14)(2,125,25,20,15)(3,121,21,16,11)(4,122,22,17,12)(5,123,23,18,13)(6,203,117,112,107)(7,204,118,113,108)(8,205,119,114,109)(9,201,120,115,110)(10,202,116,111,106)(26,175,170,165,160)(27,171,166,161,156)(28,172,167,162,157)(29,173,168,163,158)(30,174,169,164,159)(31,126,46,41,36)(32,127,47,42,37)(33,128,48,43,38)(34,129,49,44,39)(35,130,50,45,40)(51,200,195,190,185)(52,196,191,186,181)(53,197,192,187,182)(54,198,193,188,183)(55,199,194,189,184)(56,151,71,66,61)(57,152,72,67,62)(58,153,73,68,63)(59,154,74,69,64)(60,155,75,70,65)(76,225,220,215,210)(77,221,216,211,206)(78,222,217,212,207)(79,223,218,213,208)(80,224,219,214,209)(81,176,96,91,86)(82,177,97,92,87)(83,178,98,93,88)(84,179,99,94,89)(85,180,100,95,90)(101,244,239,234,229)(102,245,240,235,230)(103,241,236,231,226)(104,242,237,232,227)(105,243,238,233,228)(131,246,146,141,136)(132,247,147,142,137)(133,248,148,143,138)(134,249,149,144,139)(135,250,150,145,140), (2,5)(3,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,122)(12,121)(13,125)(14,124)(15,123)(16,22)(17,21)(18,25)(19,24)(20,23)(26,232)(27,231)(28,235)(29,234)(30,233)(36,202)(37,201)(38,205)(39,204)(40,203)(41,116)(42,120)(43,119)(44,118)(45,117)(46,111)(47,115)(48,114)(49,113)(50,112)(51,213)(52,212)(53,211)(54,215)(55,214)(56,82)(57,81)(58,85)(59,84)(60,83)(61,177)(62,176)(63,180)(64,179)(65,178)(66,97)(67,96)(68,100)(69,99)(70,98)(71,92)(72,91)(73,95)(74,94)(75,93)(76,188)(77,187)(78,186)(79,190)(80,189)(86,152)(87,151)(88,155)(89,154)(90,153)(101,163)(102,162)(103,161)(104,165)(105,164)(106,126)(107,130)(108,129)(109,128)(110,127)(131,132)(133,135)(136,247)(137,246)(138,250)(139,249)(140,248)(141,147)(142,146)(143,150)(144,149)(145,148)(156,226)(157,230)(158,229)(159,228)(160,227)(166,241)(167,245)(168,244)(169,243)(170,242)(171,236)(172,240)(173,239)(174,238)(175,237)(181,207)(182,206)(183,210)(184,209)(185,208)(191,222)(192,221)(193,225)(194,224)(195,223)(196,217)(197,216)(198,220)(199,219)(200,218) );

G=PermutationGroup([[(1,134),(2,135),(3,131),(4,132),(5,133),(6,227),(7,228),(8,229),(9,230),(10,226),(11,136),(12,137),(13,138),(14,139),(15,140),(16,141),(17,142),(18,143),(19,144),(20,145),(21,146),(22,147),(23,148),(24,149),(25,150),(26,130),(27,126),(28,127),(29,128),(30,129),(31,156),(32,157),(33,158),(34,159),(35,160),(36,161),(37,162),(38,163),(39,164),(40,165),(41,166),(42,167),(43,168),(44,169),(45,170),(46,171),(47,172),(48,173),(49,174),(50,175),(51,155),(52,151),(53,152),(54,153),(55,154),(56,181),(57,182),(58,183),(59,184),(60,185),(61,186),(62,187),(63,188),(64,189),(65,190),(66,191),(67,192),(68,193),(69,194),(70,195),(71,196),(72,197),(73,198),(74,199),(75,200),(76,180),(77,176),(78,177),(79,178),(80,179),(81,206),(82,207),(83,208),(84,209),(85,210),(86,211),(87,212),(88,213),(89,214),(90,215),(91,216),(92,217),(93,218),(94,219),(95,220),(96,221),(97,222),(98,223),(99,224),(100,225),(101,205),(102,201),(103,202),(104,203),(105,204),(106,231),(107,232),(108,233),(109,234),(110,235),(111,236),(112,237),(113,238),(114,239),(115,240),(116,241),(117,242),(118,243),(119,244),(120,245),(121,246),(122,247),(123,248),(124,249),(125,250)], [(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),(221,222,223,224,225),(226,227,228,229,230),(231,232,233,234,235),(236,237,238,239,240),(241,242,243,244,245),(246,247,248,249,250)], [(1,7,84,59,34),(2,8,85,60,35),(3,9,81,56,31),(4,10,82,57,32),(5,6,83,58,33),(11,110,86,61,36),(12,106,87,62,37),(13,107,88,63,38),(14,108,89,64,39),(15,109,90,65,40),(16,115,91,66,41),(17,111,92,67,42),(18,112,93,68,43),(19,113,94,69,44),(20,114,95,70,45),(21,120,96,71,46),(22,116,97,72,47),(23,117,98,73,48),(24,118,99,74,49),(25,119,100,75,50),(26,250,101,76,51),(27,246,102,77,52),(28,247,103,78,53),(29,248,104,79,54),(30,249,105,80,55),(121,201,176,151,126),(122,202,177,152,127),(123,203,178,153,128),(124,204,179,154,129),(125,205,180,155,130),(131,230,206,181,156),(132,226,207,182,157),(133,227,208,183,158),(134,228,209,184,159),(135,229,210,185,160),(136,235,211,186,161),(137,231,212,187,162),(138,232,213,188,163),(139,233,214,189,164),(140,234,215,190,165),(141,240,216,191,166),(142,236,217,192,167),(143,237,218,193,168),(144,238,219,194,169),(145,239,220,195,170),(146,245,221,196,171),(147,241,222,197,172),(148,242,223,198,173),(149,243,224,199,174),(150,244,225,200,175)], [(1,124,24,19,14),(2,125,25,20,15),(3,121,21,16,11),(4,122,22,17,12),(5,123,23,18,13),(6,203,117,112,107),(7,204,118,113,108),(8,205,119,114,109),(9,201,120,115,110),(10,202,116,111,106),(26,175,170,165,160),(27,171,166,161,156),(28,172,167,162,157),(29,173,168,163,158),(30,174,169,164,159),(31,126,46,41,36),(32,127,47,42,37),(33,128,48,43,38),(34,129,49,44,39),(35,130,50,45,40),(51,200,195,190,185),(52,196,191,186,181),(53,197,192,187,182),(54,198,193,188,183),(55,199,194,189,184),(56,151,71,66,61),(57,152,72,67,62),(58,153,73,68,63),(59,154,74,69,64),(60,155,75,70,65),(76,225,220,215,210),(77,221,216,211,206),(78,222,217,212,207),(79,223,218,213,208),(80,224,219,214,209),(81,176,96,91,86),(82,177,97,92,87),(83,178,98,93,88),(84,179,99,94,89),(85,180,100,95,90),(101,244,239,234,229),(102,245,240,235,230),(103,241,236,231,226),(104,242,237,232,227),(105,243,238,233,228),(131,246,146,141,136),(132,247,147,142,137),(133,248,148,143,138),(134,249,149,144,139),(135,250,150,145,140)], [(2,5),(3,4),(6,35),(7,34),(8,33),(9,32),(10,31),(11,122),(12,121),(13,125),(14,124),(15,123),(16,22),(17,21),(18,25),(19,24),(20,23),(26,232),(27,231),(28,235),(29,234),(30,233),(36,202),(37,201),(38,205),(39,204),(40,203),(41,116),(42,120),(43,119),(44,118),(45,117),(46,111),(47,115),(48,114),(49,113),(50,112),(51,213),(52,212),(53,211),(54,215),(55,214),(56,82),(57,81),(58,85),(59,84),(60,83),(61,177),(62,176),(63,180),(64,179),(65,178),(66,97),(67,96),(68,100),(69,99),(70,98),(71,92),(72,91),(73,95),(74,94),(75,93),(76,188),(77,187),(78,186),(79,190),(80,189),(86,152),(87,151),(88,155),(89,154),(90,153),(101,163),(102,162),(103,161),(104,165),(105,164),(106,126),(107,130),(108,129),(109,128),(110,127),(131,132),(133,135),(136,247),(137,246),(138,250),(139,249),(140,248),(141,147),(142,146),(143,150),(144,149),(145,148),(156,226),(157,230),(158,229),(159,228),(160,227),(166,241),(167,245),(168,244),(169,243),(170,242),(171,236),(172,240),(173,239),(174,238),(175,237),(181,207),(182,206),(183,210),(184,209),(185,208),(191,222),(192,221),(193,225),(194,224),(195,223),(196,217),(197,216),(198,220),(199,219),(200,218)]])

128 conjugacy classes

class	1	2A	2B	2C	5A	···	5BJ	10A	···	10BJ
order	1	2	2	2	5	···	5	10	···	10
size	1	1	125	125	2	···	2	2	···	2

128 irreducible representations

dim	1	1	1	2	2
type	+	+	+	+	+
image	C₁	C₂	C₂	D₅	D₁₀
kernel	C₂×C₅³⋊C₂	C₅³⋊C₂	C₅²×C₁₀	C₅×C₁₀	C₅²
# reps	1	2	1	62	62

Matrix representation of C₂×C₅³⋊C₂ ►in GL₇(𝔽₁₁)

10	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

1	0	0	0	0	0	0
0	4	10	0	0	0	0
0	5	10	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	4	4
0	0	0	0	0	7	10

1	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	10	4	0	0
0	0	0	7	4	0	0
0	0	0	0	0	0	1
0	0	0	0	0	10	7

1	0	0	0	0	0	0
0	10	1	0	0	0	0
0	6	4	0	0	0	0
0	0	0	7	1	0	0
0	0	0	10	0	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

10	0	0	0	0	0	0
0	7	1	0	0	0	0
0	7	4	0	0	0	0
0	0	0	7	4	0	0
0	0	0	10	4	0	0
0	0	0	0	0	1	0
0	0	0	0	0	7	10

G:=sub<GL(7,GF(11))| [10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,4,5,0,0,0,0,0,10,10,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,7,0,0,0,0,0,4,10],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,10,7,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,10,0,0,0,0,0,1,7],[1,0,0,0,0,0,0,0,10,6,0,0,0,0,0,1,4,0,0,0,0,0,0,0,7,10,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[10,0,0,0,0,0,0,0,7,7,0,0,0,0,0,1,4,0,0,0,0,0,0,0,7,10,0,0,0,0,0,4,4,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,10] >;

C₂×C₅³⋊C₂ in GAP, Magma, Sage, TeX

C_2\times C_5^3\rtimes C_2

% in TeX

G:=Group("C2xC5^3:C2");

// GroupNames label

G:=SmallGroup(500,55);

// by ID

G=gap.SmallGroup(500,55);

# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,242,1603,10004]);

// Polycyclic

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

// generators/relations

G = C2×C53⋊C2 order 500 = 22·53

Direct product of C2 and C53⋊C2

G = C₂×C₅³⋊C₂ order 500 = 2²·5³

Direct product of C₂ and C₅³⋊C₂