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G = C5×Dic3.D4order 480 = 25·3·5

Direct product of C5 and Dic3.D4

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

Aliases: C5×Dic3.D4, (C2×C30)⋊7Q8, C6.4(Q8×C10), C4⋊Dic32C10, (C2×C10)⋊7Dic6, C6.16(D4×C10), C30.85(C2×Q8), Dic3⋊C44C10, (C2×Dic6)⋊2C10, C30.352(C2×D4), (C2×C20).232D6, C10.168(S3×D4), C1527(C22⋊Q8), Dic3.6(C5×D4), C2.6(C10×Dic6), C222(C5×Dic6), (C10×Dic6)⋊18C2, C23.17(S3×C10), (C5×Dic3).43D4, C10.44(C2×Dic6), (C22×C10).87D6, C30.243(C4○D4), (C2×C30).398C23, (C2×C60).325C22, C6.D4.2C10, C10.107(D42S3), (C22×Dic3).3C10, (C22×C30).113C22, (C10×Dic3).214C22, (C2×C6)⋊(C5×Q8), C2.6(C5×S3×D4), C31(C5×C22⋊Q8), (C2×C4).5(S3×C10), C6.19(C5×C4○D4), (C2×C12).1(C2×C10), (C5×C4⋊Dic3)⋊20C2, C2.6(C5×D42S3), C22⋊C4.1(C5×S3), (C5×C22⋊C4).4S3, C22.39(S3×C2×C10), (C5×Dic3⋊C4)⋊20C2, (C15×C22⋊C4).6C2, (C3×C22⋊C4).1C10, (C22×C6).8(C2×C10), (Dic3×C2×C10).11C2, (C2×C6).19(C22×C10), (C2×Dic3).5(C2×C10), (C5×C6.D4).8C2, (C2×C10).332(C22×S3), SmallGroup(480,757)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C5×Dic3.D4
C1C3C6C2×C6C2×C30C10×Dic3Dic3×C2×C10 — C5×Dic3.D4
C3C2×C6 — C5×Dic3.D4
C1C2×C10C5×C22⋊C4

Generators and relations for C5×Dic3.D4
 G = < a,b,c,d,e | a5=b6=d4=e2=1, c2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, ede=b3d-1 >

Subgroups: 308 in 148 conjugacy classes, 70 normal (58 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, C20 [×7], C2×C10, C2×C10 [×2], C2×C10 [×2], Dic6 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, C2×C20 [×2], C2×C20 [×6], C5×Q8 [×2], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3 [×2], C5×Dic3 [×3], C60 [×2], C2×C30, C2×C30 [×2], C2×C30 [×2], C5×C22⋊C4, C5×C22⋊C4, C5×C4⋊C4 [×3], C22×C20, Q8×C10, Dic3.D4, C5×Dic6 [×2], C10×Dic3 [×4], C10×Dic3 [×2], C2×C60 [×2], C22×C30, C5×C22⋊Q8, C5×Dic3⋊C4 [×2], C5×C4⋊Dic3, C5×C6.D4, C15×C22⋊C4, C10×Dic6, Dic3×C2×C10, C5×Dic3.D4
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], Q8 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], Dic6 [×2], C22×S3, C5×S3, C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, C2×Dic6, S3×D4, D42S3, S3×C10 [×3], D4×C10, Q8×C10, C5×C4○D4, Dic3.D4, C5×Dic6 [×2], S3×C2×C10, C5×C22⋊Q8, C10×Dic6, C5×S3×D4, C5×D42S3, C5×Dic3.D4

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

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B5C5D6A6B6C6D6E10A···10L10M···10T12A12B12C12D15A15B15C15D20A···20H20I···20X20Y···20AF30A···30L30M···30T60A···60P
order12222234444444455556666610···1010···10121212121515151520···2020···2020···2030···3030···3060···60
size111122244666612121111222441···12···2444422224···46···612···122···24···44···4

120 irreducible representations

dim11111111111111222222222222224444
type+++++++++-++-+-
imageC1C2C2C2C2C2C2C5C10C10C10C10C10C10S3D4Q8D6D6C4○D4Dic6C5×S3C5×D4C5×Q8S3×C10S3×C10C5×C4○D4C5×Dic6S3×D4D42S3C5×S3×D4C5×D42S3
kernelC5×Dic3.D4C5×Dic3⋊C4C5×C4⋊Dic3C5×C6.D4C15×C22⋊C4C10×Dic6Dic3×C2×C10Dic3.D4Dic3⋊C4C4⋊Dic3C6.D4C3×C22⋊C4C2×Dic6C22×Dic3C5×C22⋊C4C5×Dic3C2×C30C2×C20C22×C10C30C2×C10C22⋊C4Dic3C2×C6C2×C4C23C6C22C10C10C2C2
# reps121111148444441222124488848161144

Matrix representation of C5×Dic3.D4 in GL4(𝔽61) generated by

58000
05800
00200
00020
,
1000
0100
0001
00601
,
60000
06000
003821
005923
,
06000
1000
003846
001523
,
1000
06000
0010
0001
G:=sub<GL(4,GF(61))| [58,0,0,0,0,58,0,0,0,0,20,0,0,0,0,20],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,1],[60,0,0,0,0,60,0,0,0,0,38,59,0,0,21,23],[0,1,0,0,60,0,0,0,0,0,38,15,0,0,46,23],[1,0,0,0,0,60,0,0,0,0,1,0,0,0,0,1] >;

C5×Dic3.D4 in GAP, Magma, Sage, TeX

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

G:=Group("C5xDic3.D4");
// GroupNames label

G:=SmallGroup(480,757);
// by ID

G=gap.SmallGroup(480,757);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,280,926,891,226,15686]);
// Polycyclic

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

׿
×
𝔽