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## G = C5×S3×C4⋊C4order 480 = 25·3·5

### Direct product of C5, S3 and C4⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C5×S3×C4⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C30 — S3×C2×C10 — S3×C2×C20 — C5×S3×C4⋊C4
 Lower central C3 — C6 — C5×S3×C4⋊C4
 Upper central C1 — C2×C10 — C5×C4⋊C4

Generators and relations for C5×S3×C4⋊C4
G = < a,b,c,d,e | a5=b3=c2=d4=e4=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 372 in 184 conjugacy classes, 98 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, S3 [×4], C6 [×3], C2×C4, C2×C4 [×2], C2×C4 [×11], C23, C10 [×3], C10 [×4], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C4⋊C4, C4⋊C4 [×3], C22×C4 [×3], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C4×S3 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×4], C30 [×3], C2×C4⋊C4, C2×C20, C2×C20 [×2], C2×C20 [×11], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C3×C4⋊C4, S3×C2×C4, S3×C2×C4 [×2], C5×Dic3 [×2], C5×Dic3 [×2], C60 [×2], C60 [×2], S3×C10 [×6], C2×C30, C5×C4⋊C4, C5×C4⋊C4 [×3], C22×C20 [×3], S3×C4⋊C4, S3×C20 [×4], S3×C20 [×4], C10×Dic3, C10×Dic3 [×2], C2×C60, C2×C60 [×2], S3×C2×C10, C10×C4⋊C4, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C15×C4⋊C4, S3×C2×C20, S3×C2×C20 [×2], C5×S3×C4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, C10 [×7], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C20 [×4], C2×C10 [×7], C4×S3 [×2], C22×S3, C5×S3, C2×C4⋊C4, C2×C20 [×6], C5×D4 [×2], C5×Q8 [×2], C22×C10, S3×C2×C4, S3×D4, S3×Q8, S3×C10 [×3], C5×C4⋊C4 [×4], C22×C20, D4×C10, Q8×C10, S3×C4⋊C4, S3×C20 [×2], S3×C2×C10, C10×C4⋊C4, S3×C2×C20, C5×S3×D4, C5×S3×Q8, C5×S3×C4⋊C4

Smallest permutation representation of C5×S3×C4⋊C4
On 240 points
Generators in S240
(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 90 33)(2 86 34)(3 87 35)(4 88 31)(5 89 32)(6 45 58)(7 41 59)(8 42 60)(9 43 56)(10 44 57)(11 36 193)(12 37 194)(13 38 195)(14 39 191)(15 40 192)(16 61 67)(17 62 68)(18 63 69)(19 64 70)(20 65 66)(21 233 203)(22 234 204)(23 235 205)(24 231 201)(25 232 202)(26 52 78)(27 53 79)(28 54 80)(29 55 76)(30 51 77)(46 221 228)(47 222 229)(48 223 230)(49 224 226)(50 225 227)(71 123 129)(72 124 130)(73 125 126)(74 121 127)(75 122 128)(81 150 111)(82 146 112)(83 147 113)(84 148 114)(85 149 115)(91 98 118)(92 99 119)(93 100 120)(94 96 116)(95 97 117)(101 108 136)(102 109 137)(103 110 138)(104 106 139)(105 107 140)(131 183 189)(132 184 190)(133 185 186)(134 181 187)(135 182 188)(141 210 171)(142 206 172)(143 207 173)(144 208 174)(145 209 175)(151 158 178)(152 159 179)(153 160 180)(154 156 176)(155 157 177)(161 168 196)(162 169 197)(163 170 198)(164 166 199)(165 167 200)(211 218 238)(212 219 239)(213 220 240)(214 216 236)(215 217 237)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 65)(7 61)(8 62)(9 63)(10 64)(11 211)(12 212)(13 213)(14 214)(15 215)(16 41)(17 42)(18 43)(19 44)(20 45)(21 50)(22 46)(23 47)(24 48)(25 49)(31 55)(32 51)(33 52)(34 53)(35 54)(36 238)(37 239)(38 240)(39 236)(40 237)(56 69)(57 70)(58 66)(59 67)(60 68)(71 96)(72 97)(73 98)(74 99)(75 100)(76 88)(77 89)(78 90)(79 86)(80 87)(81 103)(82 104)(83 105)(84 101)(85 102)(91 125)(92 121)(93 122)(94 123)(95 124)(106 112)(107 113)(108 114)(109 115)(110 111)(116 129)(117 130)(118 126)(119 127)(120 128)(131 156)(132 157)(133 158)(134 159)(135 160)(136 148)(137 149)(138 150)(139 146)(140 147)(141 163)(142 164)(143 165)(144 161)(145 162)(151 185)(152 181)(153 182)(154 183)(155 184)(166 172)(167 173)(168 174)(169 175)(170 171)(176 189)(177 190)(178 186)(179 187)(180 188)(191 216)(192 217)(193 218)(194 219)(195 220)(196 208)(197 209)(198 210)(199 206)(200 207)(201 223)(202 224)(203 225)(204 221)(205 222)(226 232)(227 233)(228 234)(229 235)(230 231)
(1 163 58 186)(2 164 59 187)(3 165 60 188)(4 161 56 189)(5 162 57 190)(6 133 90 170)(7 134 86 166)(8 135 87 167)(9 131 88 168)(10 132 89 169)(11 111 48 98)(12 112 49 99)(13 113 50 100)(14 114 46 96)(15 115 47 97)(16 152 53 206)(17 153 54 207)(18 154 55 208)(19 155 51 209)(20 151 52 210)(21 75 213 107)(22 71 214 108)(23 72 215 109)(24 73 211 110)(25 74 212 106)(26 141 66 178)(27 142 67 179)(28 143 68 180)(29 144 69 176)(30 145 70 177)(31 196 43 183)(32 197 44 184)(33 198 45 185)(34 199 41 181)(35 200 42 182)(36 81 223 118)(37 82 224 119)(38 83 225 120)(39 84 221 116)(40 85 222 117)(61 159 79 172)(62 160 80 173)(63 156 76 174)(64 157 77 175)(65 158 78 171)(91 193 150 230)(92 194 146 226)(93 195 147 227)(94 191 148 228)(95 192 149 229)(101 204 129 236)(102 205 130 237)(103 201 126 238)(104 202 127 239)(105 203 128 240)(121 219 139 232)(122 220 140 233)(123 216 136 234)(124 217 137 235)(125 218 138 231)
(1 126 66 81)(2 127 67 82)(3 128 68 83)(4 129 69 84)(5 130 70 85)(6 110 52 91)(7 106 53 92)(8 107 54 93)(9 108 55 94)(10 109 51 95)(11 198 231 158)(12 199 232 159)(13 200 233 160)(14 196 234 156)(15 197 235 157)(16 146 86 74)(17 147 87 75)(18 148 88 71)(19 149 89 72)(20 150 90 73)(21 153 195 167)(22 154 191 168)(23 155 192 169)(24 151 193 170)(25 152 194 166)(26 118 58 103)(27 119 59 104)(28 120 60 105)(29 116 56 101)(30 117 57 102)(31 123 63 114)(32 124 64 115)(33 125 65 111)(34 121 61 112)(35 122 62 113)(36 163 201 178)(37 164 202 179)(38 165 203 180)(39 161 204 176)(40 162 205 177)(41 139 79 99)(42 140 80 100)(43 136 76 96)(44 137 77 97)(45 138 78 98)(46 183 216 174)(47 184 217 175)(48 185 218 171)(49 181 219 172)(50 182 220 173)(131 214 208 228)(132 215 209 229)(133 211 210 230)(134 212 206 226)(135 213 207 227)(141 223 186 238)(142 224 187 239)(143 225 188 240)(144 221 189 236)(145 222 190 237)

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

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

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

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A ··· 4F 4G ··· 4L 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 12A ··· 12F 15A 15B 15C 15D 20A ··· 20X 20Y ··· 20AV 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 3 3 3 3 2 2 ··· 2 6 ··· 6 1 1 1 1 2 2 2 1 ··· 1 3 ··· 3 4 ··· 4 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2 4 ··· 4

150 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 4 4 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 S3 D4 Q8 D6 C4×S3 C5×S3 C5×D4 C5×Q8 S3×C10 S3×C20 S3×D4 S3×Q8 C5×S3×D4 C5×S3×Q8 kernel C5×S3×C4⋊C4 C5×Dic3⋊C4 C5×C4⋊Dic3 C15×C4⋊C4 S3×C2×C20 S3×C20 S3×C4⋊C4 Dic3⋊C4 C4⋊Dic3 C3×C4⋊C4 S3×C2×C4 C4×S3 C5×C4⋊C4 S3×C10 S3×C10 C2×C20 C20 C4⋊C4 D6 D6 C2×C4 C4 C10 C10 C2 C2 # reps 1 2 1 1 3 8 4 8 4 4 12 32 1 2 2 3 4 4 8 8 12 16 1 1 4 4

Matrix representation of C5×S3×C4⋊C4 in GL4(𝔽61) generated by

 34 0 0 0 0 34 0 0 0 0 1 0 0 0 0 1
,
 60 60 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 60 0 0 0 1 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 43 26 0 0 18 18
,
 50 0 0 0 0 50 0 0 0 0 60 59 0 0 0 1
G:=sub<GL(4,GF(61))| [34,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,60,0,0,0,0,0,1,0,0,0,0,1],[60,1,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,43,18,0,0,26,18],[50,0,0,0,0,50,0,0,0,0,60,0,0,0,59,1] >;

C5×S3×C4⋊C4 in GAP, Magma, Sage, TeX

C_5\times S_3\times C_4\rtimes C_4
% in TeX

G:=Group("C5xS3xC4:C4");
// GroupNames label

G:=SmallGroup(480,770);
// by ID

G=gap.SmallGroup(480,770);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,891,226,15686]);
// Polycyclic

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

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