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G = C22×D5×Dic3  order 480 = 25·3·5

Direct product of C22, D5 and Dic3

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

Aliases: C22×D5×Dic3, C30.40C24, Dic15⋊8C23, C15⋊5(C23×C4), C30⋊5(C22×C4), C5⋊2(C23×Dic3), (C23×D5).7S3, C6.40(C23×D5), C23.67(S3×D5), C10.40(S3×C23), (C5×Dic3)⋊6C23, (C6×D5).51C23, C10⋊2(C22×Dic3), (C22×C6).97D10, (C2×C30).243C23, D10.58(C22×S3), (C22×D5).114D6, (C22×C10).114D6, (C22×Dic15)⋊17C2, (C2×Dic15)⋊35C22, (C10×Dic3)⋊28C22, (C22×C30).81C22, C6⋊4(C2×C4×D5), (D5×C2×C6)⋊7C4, C3⋊4(D5×C22×C4), (C2×C6)⋊15(C4×D5), (C2×C30)⋊23(C2×C4), (C6×D5)⋊28(C2×C4), C2.2(C22×S3×D5), (Dic3×C2×C10)⋊8C2, (D5×C22×C6).3C2, (C3×D5)⋊3(C22×C4), C22.106(C2×S3×D5), (C2×C10)⋊13(C2×Dic3), (D5×C2×C6).121C22, (C2×C6).249(C22×D5), (C2×C10).249(C22×S3), SmallGroup(480,1112)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C15 — C22×D5×Dic3
Chief series
C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C22×D5×Dic3
Lower central
C15 — C22×D5×Dic3

Subgroups: 1724 in 472 conjugacy classes, 228 normal (18 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×8], C22 [×7], C22 [×28], C5, C6, C6 [×6], C6 [×8], C2×C4 [×28], C23, C23 [×14], D5 [×8], C10, C10 [×6], Dic3 [×4], Dic3 [×4], C2×C6 [×7], C2×C6 [×28], C15, C22×C4 [×14], C24, Dic5 [×4], C20 [×4], D10 [×28], C2×C10 [×7], C2×Dic3 [×6], C2×Dic3 [×22], C22×C6, C22×C6 [×14], C3×D5 [×8], C30, C30 [×6], C23×C4, C4×D5 [×16], C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×14], C22×C10, C22×Dic3, C22×Dic3 [×13], C23×C6, C5×Dic3 [×4], Dic15 [×4], C6×D5 [×28], C2×C30 [×7], C2×C4×D5 [×12], C22×Dic5, C22×C20, C23×D5, C23×Dic3, D5×Dic3 [×16], C10×Dic3 [×6], C2×Dic15 [×6], D5×C2×C6 [×14], C22×C30, D5×C22×C4, C2×D5×Dic3 [×12], Dic3×C2×C10, C22×Dic15, D5×C22×C6, C22×D5×Dic3

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D5, Dic3 [×8], D6 [×7], C22×C4 [×14], C24, D10 [×7], C2×Dic3 [×28], C22×S3 [×7], C23×C4, C4×D5 [×4], C22×D5 [×7], C22×Dic3 [×14], S3×C23, S3×D5, C2×C4×D5 [×6], C23×D5, C23×Dic3, D5×Dic3 [×4], C2×S3×D5 [×3], D5×C22×C4, C2×D5×Dic3 [×6], C22×S3×D5, C22×D5×Dic3

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c5=d2=e6=1, f2=e3, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

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

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

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

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

(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)

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

Matrix representation ►G ⊆ GL5(𝔽61)

10000
01000
00100
000600
000060
,
10000
060000
006000
00010
00001
,
10000
060100
0421800
00010
00001
,
10000
060000
042100
000600
000060
,
600000
01000
00100
0005920
00091
,
110000
01000
00100
000535
000248

G:=sub<GL(5,GF(61))| [1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,42,0,0,0,1,18,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,60,42,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,59,9,0,0,0,20,1],[11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,53,24,0,0,0,5,8] >;
 

96 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P5A5B6A···6G6H···6O10A···10N15A15B20A···20P30A···30N
order12···22···234···44···4556···66···610···10151520···2030···30
size11···15···523···315···15222···210···102···2446···64···4

96 irreducible representations

dim11111122222222444
type+++++++-+++++-+
imageC1C2C2C2C2C4S3D5Dic3D6D6D10D10C4×D5S3×D5D5×Dic3C2×S3×D5
kernelC22×D5×Dic3C2×D5×Dic3Dic3×C2×C10C22×Dic15D5×C22×C6D5×C2×C6C23×D5C22×Dic3C22×D5C22×D5C22×C10C2×Dic3C22×C6C2×C6C23C22C22
# reps112111161286112216286

In GAP, Magma, Sage, TeX 

C_2^2\times D_5\times Dic_3
 
% in TeX
 
G:=Group("C2^2xD5xDic3");
 
// GroupNames label
 
G:=SmallGroup(480,1112);
 
// by ID
 
G=gap.SmallGroup(480,1112);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,1356,18822]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^5=d^2=e^6=1,f^2=e^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
 
// generators/relations
 

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