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G = C5⋊(C423S3)  order 480 = 25·3·5

The semidirect product of C5 and C423S3 acting via C423S3/D6⋊C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D6⋊C4.1D5, C52(C423S3), (C4×Dic5)⋊10S3, (C2×C20).184D6, D6⋊Dic5.8C2, C6.Dic108C2, Dic155C49C2, (C12×Dic5)⋊21C2, C6.27(C4○D20), C30.38(C4○D4), (C2×C12).259D10, (C2×C30).62C23, C30.4Q823C2, (C22×S3).8D10, C1510(C422C2), C10.53(C4○D12), C6.39(D42D5), (C2×C60).382C22, (C2×Dic5).163D6, (C2×Dic3).17D10, C31(C23.D10), C2.10(D125D5), C2.19(D6.D10), C2.13(Dic3.D10), (C6×Dic5).185C22, (C10×Dic3).36C22, (C2×Dic15).59C22, (C5×D6⋊C4).12C2, (C2×C4).122(S3×D5), (S3×C2×C10).8C22, C22.148(C2×S3×D5), (C2×C6).74(C22×D5), (C2×C10).74(C22×S3), SmallGroup(480,448)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C5⋊(C423S3)
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — C5⋊(C423S3)
C15C2×C30 — C5⋊(C423S3)
C1C22C2×C4

Generators and relations for C5⋊(C423S3)
 G = < a,b,c,d,e | a5=b4=c4=d3=e2=1, ab=ba, cac-1=a-1, ad=da, ae=ea, bc=cb, bd=db, ebe=bc2, cd=dc, ece=b2c-1, ede=d-1 >

Subgroups: 524 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, C10 [×3], C10, Dic3 [×3], C12 [×3], D6 [×3], C2×C6, C15, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×3], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3, C30 [×3], C422C2, C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C22×C10, Dic3⋊C4 [×3], D6⋊C4, D6⋊C4 [×2], C4×C12, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60, S3×C10 [×3], C2×C30, C4×Dic5, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C5×C22⋊C4, C423S3, C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, C23.D10, D6⋊Dic5 [×2], Dic155C4, C6.Dic10, C12×Dic5, C5×D6⋊C4, C30.4Q8, C5⋊(C423S3)
Quotients: C1, C2 [×7], C22 [×7], S3, C23, D5, D6 [×3], C4○D4 [×3], D10 [×3], C22×S3, C422C2, C22×D5, C4○D12 [×3], S3×D5, C4○D20, D42D5 [×2], C423S3, C2×S3×D5, C23.D10, D6.D10, D125D5, Dic3.D10, C5⋊(C423S3)

Smallest permutation representation of C5⋊(C423S3)
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)
(1 109 49 64)(2 110 50 65)(3 106 46 61)(4 107 47 62)(5 108 48 63)(6 111 51 66)(7 112 52 67)(8 113 53 68)(9 114 54 69)(10 115 55 70)(11 116 56 71)(12 117 57 72)(13 118 58 73)(14 119 59 74)(15 120 60 75)(16 91 31 76)(17 92 32 77)(18 93 33 78)(19 94 34 79)(20 95 35 80)(21 96 36 81)(22 97 37 82)(23 98 38 83)(24 99 39 84)(25 100 40 85)(26 101 41 86)(27 102 42 87)(28 103 43 88)(29 104 44 89)(30 105 45 90)(121 226 166 181)(122 227 167 182)(123 228 168 183)(124 229 169 184)(125 230 170 185)(126 231 171 186)(127 232 172 187)(128 233 173 188)(129 234 174 189)(130 235 175 190)(131 236 176 191)(132 237 177 192)(133 238 178 193)(134 239 179 194)(135 240 180 195)(136 211 151 196)(137 212 152 197)(138 213 153 198)(139 214 154 199)(140 215 155 200)(141 216 156 201)(142 217 157 202)(143 218 158 203)(144 219 159 204)(145 220 160 205)(146 221 161 206)(147 222 162 207)(148 223 163 208)(149 224 164 209)(150 225 165 210)
(1 136 19 121)(2 140 20 125)(3 139 16 124)(4 138 17 123)(5 137 18 122)(6 144 21 129)(7 143 22 128)(8 142 23 127)(9 141 24 126)(10 145 25 130)(11 149 26 134)(12 148 27 133)(13 147 28 132)(14 146 29 131)(15 150 30 135)(31 169 46 154)(32 168 47 153)(33 167 48 152)(34 166 49 151)(35 170 50 155)(36 174 51 159)(37 173 52 158)(38 172 53 157)(39 171 54 156)(40 175 55 160)(41 179 56 164)(42 178 57 163)(43 177 58 162)(44 176 59 161)(45 180 60 165)(61 199 76 184)(62 198 77 183)(63 197 78 182)(64 196 79 181)(65 200 80 185)(66 204 81 189)(67 203 82 188)(68 202 83 187)(69 201 84 186)(70 205 85 190)(71 209 86 194)(72 208 87 193)(73 207 88 192)(74 206 89 191)(75 210 90 195)(91 229 106 214)(92 228 107 213)(93 227 108 212)(94 226 109 211)(95 230 110 215)(96 234 111 219)(97 233 112 218)(98 232 113 217)(99 231 114 216)(100 235 115 220)(101 239 116 224)(102 238 117 223)(103 237 118 222)(104 236 119 221)(105 240 120 225)
(1 14 9)(2 15 10)(3 11 6)(4 12 7)(5 13 8)(16 26 21)(17 27 22)(18 28 23)(19 29 24)(20 30 25)(31 41 36)(32 42 37)(33 43 38)(34 44 39)(35 45 40)(46 56 51)(47 57 52)(48 58 53)(49 59 54)(50 60 55)(61 71 66)(62 72 67)(63 73 68)(64 74 69)(65 75 70)(76 86 81)(77 87 82)(78 88 83)(79 89 84)(80 90 85)(91 101 96)(92 102 97)(93 103 98)(94 104 99)(95 105 100)(106 116 111)(107 117 112)(108 118 113)(109 119 114)(110 120 115)(121 131 126)(122 132 127)(123 133 128)(124 134 129)(125 135 130)(136 146 141)(137 147 142)(138 148 143)(139 149 144)(140 150 145)(151 161 156)(152 162 157)(153 163 158)(154 164 159)(155 165 160)(166 176 171)(167 177 172)(168 178 173)(169 179 174)(170 180 175)(181 191 186)(182 192 187)(183 193 188)(184 194 189)(185 195 190)(196 206 201)(197 207 202)(198 208 203)(199 209 204)(200 210 205)(211 221 216)(212 222 217)(213 223 218)(214 224 219)(215 225 220)(226 236 231)(227 237 232)(228 238 233)(229 239 234)(230 240 235)
(6 11)(7 12)(8 13)(9 14)(10 15)(21 26)(22 27)(23 28)(24 29)(25 30)(36 41)(37 42)(38 43)(39 44)(40 45)(51 56)(52 57)(53 58)(54 59)(55 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 86)(67 87)(68 88)(69 89)(70 90)(71 81)(72 82)(73 83)(74 84)(75 85)(91 106)(92 107)(93 108)(94 109)(95 110)(96 116)(97 117)(98 118)(99 119)(100 120)(101 111)(102 112)(103 113)(104 114)(105 115)(121 151)(122 152)(123 153)(124 154)(125 155)(126 161)(127 162)(128 163)(129 164)(130 165)(131 156)(132 157)(133 158)(134 159)(135 160)(136 166)(137 167)(138 168)(139 169)(140 170)(141 176)(142 177)(143 178)(144 179)(145 180)(146 171)(147 172)(148 173)(149 174)(150 175)(181 226)(182 227)(183 228)(184 229)(185 230)(186 236)(187 237)(188 238)(189 239)(190 240)(191 231)(192 232)(193 233)(194 234)(195 235)(196 211)(197 212)(198 213)(199 214)(200 215)(201 221)(202 222)(203 223)(204 224)(205 225)(206 216)(207 217)(208 218)(209 219)(210 220)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E···12L15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222234444444445566610···10101010101212121212···121515202020202020202030···3060···60
size11111222210101010126060222222···212121212222210···10444444121212124···44···4

66 irreducible representations

dim11111112222222222444444
type+++++++++++++++-+-
imageC1C2C2C2C2C2C2S3D5D6D6C4○D4D10D10D10C4○D12C4○D20S3×D5D42D5C2×S3×D5D6.D10D125D5Dic3.D10
kernelC5⋊(C423S3)D6⋊Dic5Dic155C4C6.Dic10C12×Dic5C5×D6⋊C4C30.4Q8C4×Dic5D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3C10C6C2×C4C6C22C2C2C2
# reps121111112216222128242444

Matrix representation of C5⋊(C423S3) in GL6(𝔽61)

100000
010000
001000
000100
0000171
0000600
,
60200000
010000
0053500
0048800
000010
000001
,
50370000
0110000
0011000
0001100
0000649
0000855
,
100000
010000
00592000
009100
000010
000001
,
100000
55600000
00602000
000100
000010
000001

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,17,60,0,0,0,0,1,0],[60,0,0,0,0,0,20,1,0,0,0,0,0,0,53,48,0,0,0,0,5,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,0,0,0,0,0,37,11,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,6,8,0,0,0,0,49,55],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,59,9,0,0,0,0,20,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,55,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,20,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C5⋊(C423S3) in GAP, Magma, Sage, TeX

C_5\rtimes (C_4^2\rtimes_3S_3)
% in TeX

G:=Group("C5:(C4^2:3S3)");
// GroupNames label

G:=SmallGroup(480,448);
// by ID

G=gap.SmallGroup(480,448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,219,142,1356,18822]);
// Polycyclic

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

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