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## G = C4⋊Dic5⋊S3order 480 = 25·3·5

### 3rd semidirect product of C4⋊Dic5 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C4⋊Dic5⋊S3
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic3×Dic5 — C4⋊Dic5⋊S3
 Lower central C15 — C2×C30 — C4⋊Dic5⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C4⋊Dic5⋊S3
G = < a,b,c,d,e | a4=b10=d3=e2=1, c2=b5, ab=ba, cac-1=a-1, ad=da, eae=ab5, cbc-1=ebe=b-1, bd=db, cd=dc, ece=a2c, ede=d-1 >

Subgroups: 652 in 120 conjugacy classes, 44 normal (all characteristic)
C1, C2 [×3], C2, C3, C4 [×6], C22, C22 [×3], C5, S3, C6 [×3], C2×C4, C2×C4 [×5], C23, D5, C10 [×3], Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C2×Dic3 [×2], C2×Dic3, C2×C12, C2×C12 [×2], C22×S3, D15, C30 [×3], C422C2, C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4 [×3], C3×C4⋊C4, C5×Dic3 [×2], C3×Dic5 [×2], Dic15, C60, D30 [×3], C2×C30, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4 [×3], C5×C4⋊C4, C4⋊C4⋊S3, C6×Dic5 [×2], C10×Dic3 [×2], C2×Dic15, C2×C60, C22×D15, C4⋊C4⋊D5, Dic3×Dic5, D304C4 [×2], C30.Q8, C3×C4⋊Dic5, C5×Dic3⋊C4, D303C4, C4⋊Dic5⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12, D42S3, Q83S3, S3×D5, C4○D20, D42D5, Q82D5, C4⋊C4⋊S3, C2×S3×D5, C4⋊C4⋊D5, D20⋊S3, D60⋊C2, Dic3.D10, C4⋊Dic5⋊S3

Smallest permutation representation of C4⋊Dic5⋊S3
On 240 points
Generators in S240
(1 116 56 71)(2 117 57 72)(3 118 58 73)(4 119 59 74)(5 120 60 75)(6 111 51 76)(7 112 52 77)(8 113 53 78)(9 114 54 79)(10 115 55 80)(11 82 22 100)(12 83 23 91)(13 84 24 92)(14 85 25 93)(15 86 26 94)(16 87 27 95)(17 88 28 96)(18 89 29 97)(19 90 30 98)(20 81 21 99)(31 163 228 190)(32 164 229 181)(33 165 230 182)(34 166 221 183)(35 167 222 184)(36 168 223 185)(37 169 224 186)(38 170 225 187)(39 161 226 188)(40 162 227 189)(41 128 68 101)(42 129 69 102)(43 130 70 103)(44 121 61 104)(45 122 62 105)(46 123 63 106)(47 124 64 107)(48 125 65 108)(49 126 66 109)(50 127 67 110)(131 196 176 231)(132 197 177 232)(133 198 178 233)(134 199 179 234)(135 200 180 235)(136 191 171 236)(137 192 172 237)(138 193 173 238)(139 194 174 239)(140 195 175 240)(141 206 159 214)(142 207 160 215)(143 208 151 216)(144 209 152 217)(145 210 153 218)(146 201 154 219)(147 202 155 220)(148 203 156 211)(149 204 157 212)(150 205 158 213)
(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)
(1 191 6 196)(2 200 7 195)(3 199 8 194)(4 198 9 193)(5 197 10 192)(11 33 16 38)(12 32 17 37)(13 31 18 36)(14 40 19 35)(15 39 20 34)(21 221 26 226)(22 230 27 225)(23 229 28 224)(24 228 29 223)(25 227 30 222)(41 201 46 206)(42 210 47 205)(43 209 48 204)(44 208 49 203)(45 207 50 202)(51 231 56 236)(52 240 57 235)(53 239 58 234)(54 238 59 233)(55 237 60 232)(61 216 66 211)(62 215 67 220)(63 214 68 219)(64 213 69 218)(65 212 70 217)(71 171 76 176)(72 180 77 175)(73 179 78 174)(74 178 79 173)(75 177 80 172)(81 183 86 188)(82 182 87 187)(83 181 88 186)(84 190 89 185)(85 189 90 184)(91 164 96 169)(92 163 97 168)(93 162 98 167)(94 161 99 166)(95 170 100 165)(101 154 106 159)(102 153 107 158)(103 152 108 157)(104 151 109 156)(105 160 110 155)(111 131 116 136)(112 140 117 135)(113 139 118 134)(114 138 119 133)(115 137 120 132)(121 143 126 148)(122 142 127 147)(123 141 128 146)(124 150 129 145)(125 149 130 144)
(1 27 45)(2 28 46)(3 29 47)(4 30 48)(5 21 49)(6 22 50)(7 23 41)(8 24 42)(9 25 43)(10 26 44)(11 67 51)(12 68 52)(13 69 53)(14 70 54)(15 61 55)(16 62 56)(17 63 57)(18 64 58)(19 65 59)(20 66 60)(31 218 239)(32 219 240)(33 220 231)(34 211 232)(35 212 233)(36 213 234)(37 214 235)(38 215 236)(39 216 237)(40 217 238)(71 87 105)(72 88 106)(73 89 107)(74 90 108)(75 81 109)(76 82 110)(77 83 101)(78 84 102)(79 85 103)(80 86 104)(91 128 112)(92 129 113)(93 130 114)(94 121 115)(95 122 116)(96 123 117)(97 124 118)(98 125 119)(99 126 120)(100 127 111)(131 165 147)(132 166 148)(133 167 149)(134 168 150)(135 169 141)(136 170 142)(137 161 143)(138 162 144)(139 163 145)(140 164 146)(151 172 188)(152 173 189)(153 174 190)(154 175 181)(155 176 182)(156 177 183)(157 178 184)(158 179 185)(159 180 186)(160 171 187)(191 225 207)(192 226 208)(193 227 209)(194 228 210)(195 229 201)(196 230 202)(197 221 203)(198 222 204)(199 223 205)(200 224 206)
(2 10)(3 9)(4 8)(5 7)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 70)(19 69)(20 68)(21 41)(22 50)(23 49)(24 48)(25 47)(26 46)(27 45)(28 44)(29 43)(30 42)(31 204)(32 203)(33 202)(34 201)(35 210)(36 209)(37 208)(38 207)(39 206)(40 205)(52 60)(53 59)(54 58)(55 57)(71 76)(72 75)(73 74)(77 80)(78 79)(81 106)(82 105)(83 104)(84 103)(85 102)(86 101)(87 110)(88 109)(89 108)(90 107)(91 121)(92 130)(93 129)(94 128)(95 127)(96 126)(97 125)(98 124)(99 123)(100 122)(111 116)(112 115)(113 114)(117 120)(118 119)(131 171)(132 180)(133 179)(134 178)(135 177)(136 176)(137 175)(138 174)(139 173)(140 172)(141 183)(142 182)(143 181)(144 190)(145 189)(146 188)(147 187)(148 186)(149 185)(150 184)(151 164)(152 163)(153 162)(154 161)(155 170)(156 169)(157 168)(158 167)(159 166)(160 165)(191 236)(192 235)(193 234)(194 233)(195 232)(196 231)(197 240)(198 239)(199 238)(200 237)(211 229)(212 228)(213 227)(214 226)(215 225)(216 224)(217 223)(218 222)(219 221)(220 230)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 6A 6B 6C 10A ··· 10F 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 20E ··· 20L 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 12 12 12 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 60 2 4 6 6 10 10 12 20 30 30 2 2 2 2 2 2 ··· 2 4 4 20 20 20 20 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - + + - + + + image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 C4○D4 D10 D10 C4○D12 C4○D20 D4⋊2S3 Q8⋊3S3 S3×D5 D4⋊2D5 Q8⋊2D5 C2×S3×D5 D20⋊S3 D60⋊C2 Dic3.D10 kernel C4⋊Dic5⋊S3 Dic3×Dic5 D30⋊4C4 C30.Q8 C3×C4⋊Dic5 C5×Dic3⋊C4 D30⋊3C4 C4⋊Dic5 Dic3⋊C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C10 C6 C10 C10 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 2 1 1 1 1 1 2 2 1 6 4 2 4 8 1 1 2 2 2 2 4 4 4

Matrix representation of C4⋊Dic5⋊S3 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 32 57 0 0 0 0 58 29 0 0 0 0 0 0 11 0 0 0 0 0 3 50
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 16 17 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 39 8 0 0 0 0 8 22 0 0 0 0 0 0 37 54 0 0 0 0 56 24
,
 0 60 0 0 0 0 1 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 17 18 0 0 0 0 45 44 0 0 0 0 0 0 1 0 0 0 0 0 28 60

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,32,58,0,0,0,0,57,29,0,0,0,0,0,0,11,3,0,0,0,0,0,50],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,1,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,39,8,0,0,0,0,8,22,0,0,0,0,0,0,37,56,0,0,0,0,54,24],[0,1,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,1,28,0,0,0,0,0,60] >;

C4⋊Dic5⋊S3 in GAP, Magma, Sage, TeX

C_4\rtimes {\rm Dic}_5\rtimes S_3
% in TeX

G:=Group("C4:Dic5:S3");
// GroupNames label

G:=SmallGroup(480,421);
// by ID

G=gap.SmallGroup(480,421);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,176,590,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^10=d^3=e^2=1,c^2=b^5,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,e*a*e=a*b^5,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=a^2*c,e*d*e=d^-1>;
// generators/relations

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