Copied to
clipboard

G = C5×D4.Dic3order 480 = 25·3·5

Direct product of C5 and D4.Dic3

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

Aliases: C5×D4.Dic3, C60.290C23, (C3×D4).C20, (C3×Q8).C20, D4.(C5×Dic3), C1529(C8○D4), (D4×C15).5C4, (Q8×C15).5C4, C60.185(C2×C4), C12.15(C2×C20), (C2×C20).368D6, C4.Dic38C10, (C5×D4).3Dic3, Q8.2(C5×Dic3), (C5×Q8).6Dic3, C4.5(C10×Dic3), C6.27(C22×C20), C20.55(C2×Dic3), C20.248(C22×S3), C12.42(C22×C10), (C2×C60).369C22, C30.234(C22×C4), C22.1(C10×Dic3), C10.50(C22×Dic3), C33(C5×C8○D4), (C2×C3⋊C8)⋊7C10, (C10×C3⋊C8)⋊21C2, C4.42(S3×C2×C10), C3⋊C8.13(C2×C10), (C2×C6).7(C2×C20), C4○D4.5(C5×S3), C2.8(Dic3×C2×C10), (C2×C4).58(S3×C10), (C3×C4○D4).2C10, (C5×C4○D4).10S3, (C15×C4○D4).8C2, (C5×C3⋊C8).49C22, (C2×C12).43(C2×C10), (C2×C30).131(C2×C4), (C5×C4.Dic3)⋊20C2, (C2×C10).28(C2×Dic3), SmallGroup(480,827)

Series: Derived Chief Lower central Upper central

C1C6 — C5×D4.Dic3
C1C3C6C12C60C5×C3⋊C8C10×C3⋊C8 — C5×D4.Dic3
C3C6 — C5×D4.Dic3
C1C20C5×C4○D4

Generators and relations for C5×D4.Dic3
 G = < a,b,c,d,e | a5=b4=c2=1, d6=b2, e2=b2d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 180 in 124 conjugacy classes, 90 normal (24 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, C6, C6 [×3], C8 [×4], C2×C4 [×3], D4 [×3], Q8, C10, C10 [×3], C12, C12 [×3], C2×C6 [×3], C15, C2×C8 [×3], M4(2) [×3], C4○D4, C20, C20 [×3], C2×C10 [×3], C3⋊C8, C3⋊C8 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C30, C30 [×3], C8○D4, C40 [×4], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×C3⋊C8 [×3], C4.Dic3 [×3], C3×C4○D4, C60, C60 [×3], C2×C30 [×3], C2×C40 [×3], C5×M4(2) [×3], C5×C4○D4, D4.Dic3, C5×C3⋊C8, C5×C3⋊C8 [×3], C2×C60 [×3], D4×C15 [×3], Q8×C15, C5×C8○D4, C10×C3⋊C8 [×3], C5×C4.Dic3 [×3], C15×C4○D4, C5×D4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], C23, C10 [×7], Dic3 [×4], D6 [×3], C22×C4, C20 [×4], C2×C10 [×7], C2×Dic3 [×6], C22×S3, C5×S3, C8○D4, C2×C20 [×6], C22×C10, C22×Dic3, C5×Dic3 [×4], S3×C10 [×3], C22×C20, D4.Dic3, C10×Dic3 [×6], S3×C2×C10, C5×C8○D4, Dic3×C2×C10, C5×D4.Dic3

Smallest permutation representation of C5×D4.Dic3
On 240 points
Generators in S240
(1 55 43 31 19)(2 56 44 32 20)(3 57 45 33 21)(4 58 46 34 22)(5 59 47 35 23)(6 60 48 36 24)(7 49 37 25 13)(8 50 38 26 14)(9 51 39 27 15)(10 52 40 28 16)(11 53 41 29 17)(12 54 42 30 18)(61 109 97 85 73)(62 110 98 86 74)(63 111 99 87 75)(64 112 100 88 76)(65 113 101 89 77)(66 114 102 90 78)(67 115 103 91 79)(68 116 104 92 80)(69 117 105 93 81)(70 118 106 94 82)(71 119 107 95 83)(72 120 108 96 84)(121 169 157 145 133)(122 170 158 146 134)(123 171 159 147 135)(124 172 160 148 136)(125 173 161 149 137)(126 174 162 150 138)(127 175 163 151 139)(128 176 164 152 140)(129 177 165 153 141)(130 178 166 154 142)(131 179 167 155 143)(132 180 168 156 144)(181 229 217 205 193)(182 230 218 206 194)(183 231 219 207 195)(184 232 220 208 196)(185 233 221 209 197)(186 234 222 210 198)(187 235 223 211 199)(188 236 224 212 200)(189 237 225 213 201)(190 238 226 214 202)(191 239 227 215 203)(192 240 228 216 204)
(1 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 34 31 28)(26 35 32 29)(27 36 33 30)(37 46 43 40)(38 47 44 41)(39 48 45 42)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)(97 100 103 106)(98 101 104 107)(99 102 105 108)(109 112 115 118)(110 113 116 119)(111 114 117 120)(121 130 127 124)(122 131 128 125)(123 132 129 126)(133 142 139 136)(134 143 140 137)(135 144 141 138)(145 154 151 148)(146 155 152 149)(147 156 153 150)(157 166 163 160)(158 167 164 161)(159 168 165 162)(169 178 175 172)(170 179 176 173)(171 180 177 174)(181 184 187 190)(182 185 188 191)(183 186 189 192)(193 196 199 202)(194 197 200 203)(195 198 201 204)(205 208 211 214)(206 209 212 215)(207 210 213 216)(217 220 223 226)(218 221 224 227)(219 222 225 228)(229 232 235 238)(230 233 236 239)(231 234 237 240)
(1 64)(2 65)(3 66)(4 67)(5 68)(6 69)(7 70)(8 71)(9 72)(10 61)(11 62)(12 63)(13 82)(14 83)(15 84)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 94)(26 95)(27 96)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 106)(38 107)(39 108)(40 97)(41 98)(42 99)(43 100)(44 101)(45 102)(46 103)(47 104)(48 105)(49 118)(50 119)(51 120)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 116)(60 117)(121 190)(122 191)(123 192)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 202)(134 203)(135 204)(136 193)(137 194)(138 195)(139 196)(140 197)(141 198)(142 199)(143 200)(144 201)(145 214)(146 215)(147 216)(148 205)(149 206)(150 207)(151 208)(152 209)(153 210)(154 211)(155 212)(156 213)(157 226)(158 227)(159 228)(160 217)(161 218)(162 219)(163 220)(164 221)(165 222)(166 223)(167 224)(168 225)(169 238)(170 239)(171 240)(172 229)(173 230)(174 231)(175 232)(176 233)(177 234)(178 235)(179 236)(180 237)
(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)
(1 128 10 125 7 122 4 131)(2 121 11 130 8 127 5 124)(3 126 12 123 9 132 6 129)(13 134 22 143 19 140 16 137)(14 139 23 136 20 133 17 142)(15 144 24 141 21 138 18 135)(25 146 34 155 31 152 28 149)(26 151 35 148 32 145 29 154)(27 156 36 153 33 150 30 147)(37 158 46 167 43 164 40 161)(38 163 47 160 44 157 41 166)(39 168 48 165 45 162 42 159)(49 170 58 179 55 176 52 173)(50 175 59 172 56 169 53 178)(51 180 60 177 57 174 54 171)(61 182 70 191 67 188 64 185)(62 187 71 184 68 181 65 190)(63 192 72 189 69 186 66 183)(73 194 82 203 79 200 76 197)(74 199 83 196 80 193 77 202)(75 204 84 201 81 198 78 195)(85 206 94 215 91 212 88 209)(86 211 95 208 92 205 89 214)(87 216 96 213 93 210 90 207)(97 218 106 227 103 224 100 221)(98 223 107 220 104 217 101 226)(99 228 108 225 105 222 102 219)(109 230 118 239 115 236 112 233)(110 235 119 232 116 229 113 238)(111 240 120 237 117 234 114 231)

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

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

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

150 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B5C5D6A6B6C6D8A8B8C8D8E···8J10A10B10C10D10E···10P12A12B12C12D12E15A15B15C15D20A···20H20I···20T30A30B30C30D30E···30P40A···40P40Q···40AN60A···60H60I···60T
order122223444445555666688888···81010101010···1012121212121515151520···2020···203030303030···3040···4040···4060···6060···60
size112222112221111244433336···611112···22244422221···12···222224···43···36···62···24···4

150 irreducible representations

dim111111111111222222222244
type++++++--
imageC1C2C2C2C4C4C5C10C10C10C20C20S3D6Dic3Dic3C5×S3C8○D4S3×C10C5×Dic3C5×Dic3C5×C8○D4D4.Dic3C5×D4.Dic3
kernelC5×D4.Dic3C10×C3⋊C8C5×C4.Dic3C15×C4○D4D4×C15Q8×C15D4.Dic3C2×C3⋊C8C4.Dic3C3×C4○D4C3×D4C3×Q8C5×C4○D4C2×C20C5×D4C5×Q8C4○D4C15C2×C4D4Q8C3C5C1
# reps133162412124248133144121241628

Matrix representation of C5×D4.Dic3 in GL4(𝔽241) generated by

87000
08700
00910
00091
,
240000
024000
001770
0011864
,
1000
0100
00177192
0011864
,
1100
240000
001770
000177
,
1417600
17610000
00300
00030
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,91,0,0,0,0,91],[240,0,0,0,0,240,0,0,0,0,177,118,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,177,118,0,0,192,64],[1,240,0,0,1,0,0,0,0,0,177,0,0,0,0,177],[141,176,0,0,76,100,0,0,0,0,30,0,0,0,0,30] >;

C5×D4.Dic3 in GAP, Magma, Sage, TeX

C_5\times D_4.{\rm Dic}_3
% in TeX

G:=Group("C5xD4.Dic3");
// GroupNames label

G:=SmallGroup(480,827);
// by ID

G=gap.SmallGroup(480,827);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,891,102,15686]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^4=c^2=1,d^6=b^2,e^2=b^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations

׿
×
𝔽