direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C22.D12, D6⋊C4⋊6C10, C6.6(D4×C10), C4⋊Dic3⋊5C10, (C2×C30).91D4, C2.8(C10×D12), C30.293(C2×D4), C10.77(C2×D12), (C2×C20).235D6, (C2×C10).26D12, C22.4(C5×D12), C23.21(S3×C10), (C22×C10).91D6, C30.247(C4○D4), (C2×C60).330C22, (C2×C30).406C23, (C22×Dic3)⋊2C10, C15⋊30(C22.D4), C10.112(D4⋊2S3), (C22×C30).121C22, (C10×Dic3).141C22, (C2×C6).4(C5×D4), (C5×D6⋊C4)⋊22C2, C22⋊C4⋊6(C5×S3), (C2×C4).8(S3×C10), C6.22(C5×C4○D4), (C5×C22⋊C4)⋊14S3, (C3×C22⋊C4)⋊4C10, (C2×C12).4(C2×C10), (C5×C4⋊Dic3)⋊23C2, (Dic3×C2×C10)⋊13C2, (C2×C3⋊D4).5C10, C22.45(S3×C2×C10), (C15×C22⋊C4)⋊18C2, C3⋊2(C5×C22.D4), C2.10(C5×D4⋊2S3), (C10×C3⋊D4).12C2, (S3×C2×C10).67C22, (C22×S3).6(C2×C10), (C2×C6).27(C22×C10), (C22×C6).16(C2×C10), (C2×Dic3).8(C2×C10), (C2×C10).340(C22×S3), SmallGroup(480,765)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.D12
G = < a,b,c,d,e | a5=b2=c2=d12=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=cd-1 >
Subgroups: 372 in 156 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×3], C3, C4 [×5], C22, C22 [×2], C22 [×5], C5, S3, C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×5], D4 [×2], C23, C23, C10, C10 [×2], C10 [×3], Dic3 [×3], C12 [×2], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×D4, C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×2], C3⋊D4 [×2], C2×C12 [×2], C22×S3, C22×C6, C5×S3, C30, C30 [×2], C30 [×2], C22.D4, C2×C20 [×2], C2×C20 [×5], C5×D4 [×2], C22×C10, C22×C10, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3 [×3], C60 [×2], S3×C10 [×3], C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C22×C20, D4×C10, C22.D12, C10×Dic3, C10×Dic3 [×2], C10×Dic3 [×2], C5×C3⋊D4 [×2], C2×C60 [×2], S3×C2×C10, C22×C30, C5×C22.D4, C5×C4⋊Dic3 [×2], C5×D6⋊C4 [×2], C15×C22⋊C4, Dic3×C2×C10, C10×C3⋊D4, C5×C22.D12
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C4○D4 [×2], C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C22.D4, C5×D4 [×2], C22×C10, C2×D12, D4⋊2S3 [×2], S3×C10 [×3], D4×C10, C5×C4○D4 [×2], C22.D12, C5×D12 [×2], S3×C2×C10, C5×C22.D4, C10×D12, C5×D4⋊2S3 [×2], C5×C22.D12
►On 240 points
Generators in S240
(1 75 196 100 124)(2 76 197 101 125)(3 77 198 102 126)(4 78 199 103 127)(5 79 200 104 128)(6 80 201 105 129)(7 81 202 106 130)(8 82 203 107 131)(9 83 204 108 132)(10 84 193 97 121)(11 73 194 98 122)(12 74 195 99 123)(13 162 53 71 188)(14 163 54 72 189)(15 164 55 61 190)(16 165 56 62 191)(17 166 57 63 192)(18 167 58 64 181)(19 168 59 65 182)(20 157 60 66 183)(21 158 49 67 184)(22 159 50 68 185)(23 160 51 69 186)(24 161 52 70 187)(25 47 144 110 154)(26 48 133 111 155)(27 37 134 112 156)(28 38 135 113 145)(29 39 136 114 146)(30 40 137 115 147)(31 41 138 116 148)(32 42 139 117 149)(33 43 140 118 150)(34 44 141 119 151)(35 45 142 120 152)(36 46 143 109 153)(85 219 176 229 207)(86 220 177 230 208)(87 221 178 231 209)(88 222 179 232 210)(89 223 180 233 211)(90 224 169 234 212)(91 225 170 235 213)(92 226 171 236 214)(93 227 172 237 215)(94 228 173 238 216)(95 217 174 239 205)(96 218 175 240 206)
(2 227)(4 217)(6 219)(8 221)(10 223)(12 225)(14 151)(16 153)(18 155)(20 145)(22 147)(24 149)(26 167)(28 157)(30 159)(32 161)(34 163)(36 165)(38 60)(40 50)(42 52)(44 54)(46 56)(48 58)(62 143)(64 133)(66 135)(68 137)(70 139)(72 141)(74 170)(76 172)(78 174)(80 176)(82 178)(84 180)(85 129)(87 131)(89 121)(91 123)(93 125)(95 127)(97 211)(99 213)(101 215)(103 205)(105 207)(107 209)(109 191)(111 181)(113 183)(115 185)(117 187)(119 189)(193 233)(195 235)(197 237)(199 239)(201 229)(203 231)
(1 226)(2 227)(3 228)(4 217)(5 218)(6 219)(7 220)(8 221)(9 222)(10 223)(11 224)(12 225)(13 150)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 145)(21 146)(22 147)(23 148)(24 149)(25 166)(26 167)(27 168)(28 157)(29 158)(30 159)(31 160)(32 161)(33 162)(34 163)(35 164)(36 165)(37 59)(38 60)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(61 142)(62 143)(63 144)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)(73 169)(74 170)(75 171)(76 172)(77 173)(78 174)(79 175)(80 176)(81 177)(82 178)(83 179)(84 180)(85 129)(86 130)(87 131)(88 132)(89 121)(90 122)(91 123)(92 124)(93 125)(94 126)(95 127)(96 128)(97 211)(98 212)(99 213)(100 214)(101 215)(102 216)(103 205)(104 206)(105 207)(106 208)(107 209)(108 210)(109 191)(110 192)(111 181)(112 182)(113 183)(114 184)(115 185)(116 186)(117 187)(118 188)(119 189)(120 190)(193 233)(194 234)(195 235)(196 236)(197 237)(198 238)(199 239)(200 240)(201 229)(202 230)(203 231)(204 232)
(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)
(1 116 226 186)(2 185 227 115)(3 114 228 184)(4 183 217 113)(5 112 218 182)(6 181 219 111)(7 110 220 192)(8 191 221 109)(9 120 222 190)(10 189 223 119)(11 118 224 188)(12 187 225 117)(13 73 150 169)(14 180 151 84)(15 83 152 179)(16 178 153 82)(17 81 154 177)(18 176 155 80)(19 79 156 175)(20 174 145 78)(21 77 146 173)(22 172 147 76)(23 75 148 171)(24 170 149 74)(25 230 166 202)(26 201 167 229)(27 240 168 200)(28 199 157 239)(29 238 158 198)(30 197 159 237)(31 236 160 196)(32 195 161 235)(33 234 162 194)(34 193 163 233)(35 232 164 204)(36 203 165 231)(37 206 59 104)(38 103 60 205)(39 216 49 102)(40 101 50 215)(41 214 51 100)(42 99 52 213)(43 212 53 98)(44 97 54 211)(45 210 55 108)(46 107 56 209)(47 208 57 106)(48 105 58 207)(61 132 142 88)(62 87 143 131)(63 130 144 86)(64 85 133 129)(65 128 134 96)(66 95 135 127)(67 126 136 94)(68 93 137 125)(69 124 138 92)(70 91 139 123)(71 122 140 90)(72 89 141 121)
G:=sub<Sym(240)| (1,75,196,100,124)(2,76,197,101,125)(3,77,198,102,126)(4,78,199,103,127)(5,79,200,104,128)(6,80,201,105,129)(7,81,202,106,130)(8,82,203,107,131)(9,83,204,108,132)(10,84,193,97,121)(11,73,194,98,122)(12,74,195,99,123)(13,162,53,71,188)(14,163,54,72,189)(15,164,55,61,190)(16,165,56,62,191)(17,166,57,63,192)(18,167,58,64,181)(19,168,59,65,182)(20,157,60,66,183)(21,158,49,67,184)(22,159,50,68,185)(23,160,51,69,186)(24,161,52,70,187)(25,47,144,110,154)(26,48,133,111,155)(27,37,134,112,156)(28,38,135,113,145)(29,39,136,114,146)(30,40,137,115,147)(31,41,138,116,148)(32,42,139,117,149)(33,43,140,118,150)(34,44,141,119,151)(35,45,142,120,152)(36,46,143,109,153)(85,219,176,229,207)(86,220,177,230,208)(87,221,178,231,209)(88,222,179,232,210)(89,223,180,233,211)(90,224,169,234,212)(91,225,170,235,213)(92,226,171,236,214)(93,227,172,237,215)(94,228,173,238,216)(95,217,174,239,205)(96,218,175,240,206), (2,227)(4,217)(6,219)(8,221)(10,223)(12,225)(14,151)(16,153)(18,155)(20,145)(22,147)(24,149)(26,167)(28,157)(30,159)(32,161)(34,163)(36,165)(38,60)(40,50)(42,52)(44,54)(46,56)(48,58)(62,143)(64,133)(66,135)(68,137)(70,139)(72,141)(74,170)(76,172)(78,174)(80,176)(82,178)(84,180)(85,129)(87,131)(89,121)(91,123)(93,125)(95,127)(97,211)(99,213)(101,215)(103,205)(105,207)(107,209)(109,191)(111,181)(113,183)(115,185)(117,187)(119,189)(193,233)(195,235)(197,237)(199,239)(201,229)(203,231), (1,226)(2,227)(3,228)(4,217)(5,218)(6,219)(7,220)(8,221)(9,222)(10,223)(11,224)(12,225)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,145)(21,146)(22,147)(23,148)(24,149)(25,166)(26,167)(27,168)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(61,142)(62,143)(63,144)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,129)(86,130)(87,131)(88,132)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,211)(98,212)(99,213)(100,214)(101,215)(102,216)(103,205)(104,206)(105,207)(106,208)(107,209)(108,210)(109,191)(110,192)(111,181)(112,182)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)(201,229)(202,230)(203,231)(204,232), (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), (1,116,226,186)(2,185,227,115)(3,114,228,184)(4,183,217,113)(5,112,218,182)(6,181,219,111)(7,110,220,192)(8,191,221,109)(9,120,222,190)(10,189,223,119)(11,118,224,188)(12,187,225,117)(13,73,150,169)(14,180,151,84)(15,83,152,179)(16,178,153,82)(17,81,154,177)(18,176,155,80)(19,79,156,175)(20,174,145,78)(21,77,146,173)(22,172,147,76)(23,75,148,171)(24,170,149,74)(25,230,166,202)(26,201,167,229)(27,240,168,200)(28,199,157,239)(29,238,158,198)(30,197,159,237)(31,236,160,196)(32,195,161,235)(33,234,162,194)(34,193,163,233)(35,232,164,204)(36,203,165,231)(37,206,59,104)(38,103,60,205)(39,216,49,102)(40,101,50,215)(41,214,51,100)(42,99,52,213)(43,212,53,98)(44,97,54,211)(45,210,55,108)(46,107,56,209)(47,208,57,106)(48,105,58,207)(61,132,142,88)(62,87,143,131)(63,130,144,86)(64,85,133,129)(65,128,134,96)(66,95,135,127)(67,126,136,94)(68,93,137,125)(69,124,138,92)(70,91,139,123)(71,122,140,90)(72,89,141,121)>;
G:=Group( (1,75,196,100,124)(2,76,197,101,125)(3,77,198,102,126)(4,78,199,103,127)(5,79,200,104,128)(6,80,201,105,129)(7,81,202,106,130)(8,82,203,107,131)(9,83,204,108,132)(10,84,193,97,121)(11,73,194,98,122)(12,74,195,99,123)(13,162,53,71,188)(14,163,54,72,189)(15,164,55,61,190)(16,165,56,62,191)(17,166,57,63,192)(18,167,58,64,181)(19,168,59,65,182)(20,157,60,66,183)(21,158,49,67,184)(22,159,50,68,185)(23,160,51,69,186)(24,161,52,70,187)(25,47,144,110,154)(26,48,133,111,155)(27,37,134,112,156)(28,38,135,113,145)(29,39,136,114,146)(30,40,137,115,147)(31,41,138,116,148)(32,42,139,117,149)(33,43,140,118,150)(34,44,141,119,151)(35,45,142,120,152)(36,46,143,109,153)(85,219,176,229,207)(86,220,177,230,208)(87,221,178,231,209)(88,222,179,232,210)(89,223,180,233,211)(90,224,169,234,212)(91,225,170,235,213)(92,226,171,236,214)(93,227,172,237,215)(94,228,173,238,216)(95,217,174,239,205)(96,218,175,240,206), (2,227)(4,217)(6,219)(8,221)(10,223)(12,225)(14,151)(16,153)(18,155)(20,145)(22,147)(24,149)(26,167)(28,157)(30,159)(32,161)(34,163)(36,165)(38,60)(40,50)(42,52)(44,54)(46,56)(48,58)(62,143)(64,133)(66,135)(68,137)(70,139)(72,141)(74,170)(76,172)(78,174)(80,176)(82,178)(84,180)(85,129)(87,131)(89,121)(91,123)(93,125)(95,127)(97,211)(99,213)(101,215)(103,205)(105,207)(107,209)(109,191)(111,181)(113,183)(115,185)(117,187)(119,189)(193,233)(195,235)(197,237)(199,239)(201,229)(203,231), (1,226)(2,227)(3,228)(4,217)(5,218)(6,219)(7,220)(8,221)(9,222)(10,223)(11,224)(12,225)(13,150)(14,151)(15,152)(16,153)(17,154)(18,155)(19,156)(20,145)(21,146)(22,147)(23,148)(24,149)(25,166)(26,167)(27,168)(28,157)(29,158)(30,159)(31,160)(32,161)(33,162)(34,163)(35,164)(36,165)(37,59)(38,60)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(61,142)(62,143)(63,144)(64,133)(65,134)(66,135)(67,136)(68,137)(69,138)(70,139)(71,140)(72,141)(73,169)(74,170)(75,171)(76,172)(77,173)(78,174)(79,175)(80,176)(81,177)(82,178)(83,179)(84,180)(85,129)(86,130)(87,131)(88,132)(89,121)(90,122)(91,123)(92,124)(93,125)(94,126)(95,127)(96,128)(97,211)(98,212)(99,213)(100,214)(101,215)(102,216)(103,205)(104,206)(105,207)(106,208)(107,209)(108,210)(109,191)(110,192)(111,181)(112,182)(113,183)(114,184)(115,185)(116,186)(117,187)(118,188)(119,189)(120,190)(193,233)(194,234)(195,235)(196,236)(197,237)(198,238)(199,239)(200,240)(201,229)(202,230)(203,231)(204,232), (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), (1,116,226,186)(2,185,227,115)(3,114,228,184)(4,183,217,113)(5,112,218,182)(6,181,219,111)(7,110,220,192)(8,191,221,109)(9,120,222,190)(10,189,223,119)(11,118,224,188)(12,187,225,117)(13,73,150,169)(14,180,151,84)(15,83,152,179)(16,178,153,82)(17,81,154,177)(18,176,155,80)(19,79,156,175)(20,174,145,78)(21,77,146,173)(22,172,147,76)(23,75,148,171)(24,170,149,74)(25,230,166,202)(26,201,167,229)(27,240,168,200)(28,199,157,239)(29,238,158,198)(30,197,159,237)(31,236,160,196)(32,195,161,235)(33,234,162,194)(34,193,163,233)(35,232,164,204)(36,203,165,231)(37,206,59,104)(38,103,60,205)(39,216,49,102)(40,101,50,215)(41,214,51,100)(42,99,52,213)(43,212,53,98)(44,97,54,211)(45,210,55,108)(46,107,56,209)(47,208,57,106)(48,105,58,207)(61,132,142,88)(62,87,143,131)(63,130,144,86)(64,85,133,129)(65,128,134,96)(66,95,135,127)(67,126,136,94)(68,93,137,125)(69,124,138,92)(70,91,139,123)(71,122,140,90)(72,89,141,121) );
G=PermutationGroup([(1,75,196,100,124),(2,76,197,101,125),(3,77,198,102,126),(4,78,199,103,127),(5,79,200,104,128),(6,80,201,105,129),(7,81,202,106,130),(8,82,203,107,131),(9,83,204,108,132),(10,84,193,97,121),(11,73,194,98,122),(12,74,195,99,123),(13,162,53,71,188),(14,163,54,72,189),(15,164,55,61,190),(16,165,56,62,191),(17,166,57,63,192),(18,167,58,64,181),(19,168,59,65,182),(20,157,60,66,183),(21,158,49,67,184),(22,159,50,68,185),(23,160,51,69,186),(24,161,52,70,187),(25,47,144,110,154),(26,48,133,111,155),(27,37,134,112,156),(28,38,135,113,145),(29,39,136,114,146),(30,40,137,115,147),(31,41,138,116,148),(32,42,139,117,149),(33,43,140,118,150),(34,44,141,119,151),(35,45,142,120,152),(36,46,143,109,153),(85,219,176,229,207),(86,220,177,230,208),(87,221,178,231,209),(88,222,179,232,210),(89,223,180,233,211),(90,224,169,234,212),(91,225,170,235,213),(92,226,171,236,214),(93,227,172,237,215),(94,228,173,238,216),(95,217,174,239,205),(96,218,175,240,206)], [(2,227),(4,217),(6,219),(8,221),(10,223),(12,225),(14,151),(16,153),(18,155),(20,145),(22,147),(24,149),(26,167),(28,157),(30,159),(32,161),(34,163),(36,165),(38,60),(40,50),(42,52),(44,54),(46,56),(48,58),(62,143),(64,133),(66,135),(68,137),(70,139),(72,141),(74,170),(76,172),(78,174),(80,176),(82,178),(84,180),(85,129),(87,131),(89,121),(91,123),(93,125),(95,127),(97,211),(99,213),(101,215),(103,205),(105,207),(107,209),(109,191),(111,181),(113,183),(115,185),(117,187),(119,189),(193,233),(195,235),(197,237),(199,239),(201,229),(203,231)], [(1,226),(2,227),(3,228),(4,217),(5,218),(6,219),(7,220),(8,221),(9,222),(10,223),(11,224),(12,225),(13,150),(14,151),(15,152),(16,153),(17,154),(18,155),(19,156),(20,145),(21,146),(22,147),(23,148),(24,149),(25,166),(26,167),(27,168),(28,157),(29,158),(30,159),(31,160),(32,161),(33,162),(34,163),(35,164),(36,165),(37,59),(38,60),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(61,142),(62,143),(63,144),(64,133),(65,134),(66,135),(67,136),(68,137),(69,138),(70,139),(71,140),(72,141),(73,169),(74,170),(75,171),(76,172),(77,173),(78,174),(79,175),(80,176),(81,177),(82,178),(83,179),(84,180),(85,129),(86,130),(87,131),(88,132),(89,121),(90,122),(91,123),(92,124),(93,125),(94,126),(95,127),(96,128),(97,211),(98,212),(99,213),(100,214),(101,215),(102,216),(103,205),(104,206),(105,207),(106,208),(107,209),(108,210),(109,191),(110,192),(111,181),(112,182),(113,183),(114,184),(115,185),(116,186),(117,187),(118,188),(119,189),(120,190),(193,233),(194,234),(195,235),(196,236),(197,237),(198,238),(199,239),(200,240),(201,229),(202,230),(203,231),(204,232)], [(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)], [(1,116,226,186),(2,185,227,115),(3,114,228,184),(4,183,217,113),(5,112,218,182),(6,181,219,111),(7,110,220,192),(8,191,221,109),(9,120,222,190),(10,189,223,119),(11,118,224,188),(12,187,225,117),(13,73,150,169),(14,180,151,84),(15,83,152,179),(16,178,153,82),(17,81,154,177),(18,176,155,80),(19,79,156,175),(20,174,145,78),(21,77,146,173),(22,172,147,76),(23,75,148,171),(24,170,149,74),(25,230,166,202),(26,201,167,229),(27,240,168,200),(28,199,157,239),(29,238,158,198),(30,197,159,237),(31,236,160,196),(32,195,161,235),(33,234,162,194),(34,193,163,233),(35,232,164,204),(36,203,165,231),(37,206,59,104),(38,103,60,205),(39,216,49,102),(40,101,50,215),(41,214,51,100),(42,99,52,213),(43,212,53,98),(44,97,54,211),(45,210,55,108),(46,107,56,209),(47,208,57,106),(48,105,58,207),(61,132,142,88),(62,87,143,131),(63,130,144,86),(64,85,133,129),(65,128,134,96),(66,95,135,127),(67,126,136,94),(68,93,137,125),(69,124,138,92),(70,91,139,123),(71,122,140,90),(72,89,141,121)])
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10L | 10M | ··· | 10T | 10U | 10V | 10W | 10X | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20X | 20Y | 20Z | 20AA | 20AB | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | S3 | D4 | D6 | D6 | C4○D4 | D12 | C5×S3 | C5×D4 | S3×C10 | S3×C10 | C5×C4○D4 | C5×D12 | D4⋊2S3 | C5×D4⋊2S3 |
| kernel | C5×C22.D12 | C5×C4⋊Dic3 | C5×D6⋊C4 | C15×C22⋊C4 | Dic3×C2×C10 | C10×C3⋊D4 | C22.D12 | C4⋊Dic3 | D6⋊C4 | C3×C22⋊C4 | C22×Dic3 | C2×C3⋊D4 | C5×C22⋊C4 | C2×C30 | C2×C20 | C22×C10 | C30 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C6 | C22 | C10 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 8 | 8 | 4 | 4 | 4 | 1 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 8 | 4 | 16 | 16 | 2 | 8 |
Matrix representation of C5×C22.D12 ►in GL6(𝔽61)
| 34 | 0 | 0 | 0 | 0 | 0 |
| 0 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 34 | 2 | 0 | 0 | 0 | 0 |
| 2 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 38 | 38 |
| 0 | 0 | 0 | 0 | 15 | 23 |
G:=sub<GL(6,GF(61))| [34,0,0,0,0,0,0,34,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,60,0],[34,2,0,0,0,0,2,27,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,38,15,0,0,0,0,38,23] >;
C5×C22.D12 in GAP, Magma, Sage, TeX
C_5\times C_2^2.D_{12} % in TeX
G:=Group("C5xC2^2.D12"); // GroupNames label
G:=SmallGroup(480,765);
// by ID
G=gap.SmallGroup(480,765);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,926,891,646,15686]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^12=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^-1>;
// generators/relations