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G = C6×D4.D5order 480 = 25·3·5

Direct product of C6 and D4.D5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C6×D4.D5
 Chief series C1 — C5 — C10 — C20 — C60 — C3×Dic10 — C6×Dic10 — C6×D4.D5
 Lower central C5 — C10 — C20 — C6×D4.D5
 Upper central C1 — C2×C6 — C2×C12 — C6×D4

Generators and relations for C6×D4.D5
G = < a,b,c,d,e | a6=b4=c2=d5=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 368 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×3], C23, C10, C10 [×2], C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×3], C22×C6, C30, C30 [×2], C30 [×2], C2×SD16, C52C8 [×2], Dic10 [×2], Dic10, C2×Dic5, C2×C20, C5×D4 [×2], C5×D4, C22×C10, C2×C24, C3×SD16 [×4], C6×D4, C6×Q8, C3×Dic5 [×2], C60 [×2], C2×C30, C2×C30 [×4], C2×C52C8, D4.D5 [×4], C2×Dic10, D4×C10, C6×SD16, C3×C52C8 [×2], C3×Dic10 [×2], C3×Dic10, C6×Dic5, C2×C60, D4×C15 [×2], D4×C15, C22×C30, C2×D4.D5, C6×C52C8, C3×D4.D5 [×4], C6×Dic10, D4×C30, C6×D4.D5
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], SD16 [×2], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C2×SD16, C5⋊D4 [×2], C22×D5, C3×SD16 [×2], C6×D4, C6×D5 [×3], D4.D5 [×2], C2×C5⋊D4, C6×SD16, C3×C5⋊D4 [×2], D5×C2×C6, C2×D4.D5, C3×D4.D5 [×2], C6×C5⋊D4, C6×D4.D5

Smallest permutation representation of C6×D4.D5
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 7 111 170)(2 8 112 171)(3 9 113 172)(4 10 114 173)(5 11 109 174)(6 12 110 169)(13 168 49 77)(14 163 50 78)(15 164 51 73)(16 165 52 74)(17 166 53 75)(18 167 54 76)(19 141 195 220)(20 142 196 221)(21 143 197 222)(22 144 198 217)(23 139 193 218)(24 140 194 219)(25 96 117 187)(26 91 118 188)(27 92 119 189)(28 93 120 190)(29 94 115 191)(30 95 116 192)(31 43 70 122)(32 44 71 123)(33 45 72 124)(34 46 67 125)(35 47 68 126)(36 48 69 121)(37 226 131 153)(38 227 132 154)(39 228 127 155)(40 223 128 156)(41 224 129 151)(42 225 130 152)(55 89 104 186)(56 90 105 181)(57 85 106 182)(58 86 107 183)(59 87 108 184)(60 88 103 185)(61 204 159 207)(62 199 160 208)(63 200 161 209)(64 201 162 210)(65 202 157 205)(66 203 158 206)(79 236 176 216)(80 237 177 211)(81 238 178 212)(82 239 179 213)(83 240 180 214)(84 235 175 215)(97 135 233 148)(98 136 234 149)(99 137 229 150)(100 138 230 145)(101 133 231 146)(102 134 232 147)
(1 170)(2 171)(3 172)(4 173)(5 174)(6 169)(7 111)(8 112)(9 113)(10 114)(11 109)(12 110)(13 77)(14 78)(15 73)(16 74)(17 75)(18 76)(19 195)(20 196)(21 197)(22 198)(23 193)(24 194)(25 187)(26 188)(27 189)(28 190)(29 191)(30 192)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 131)(38 132)(39 127)(40 128)(41 129)(42 130)(49 168)(50 163)(51 164)(52 165)(53 166)(54 167)(55 89)(56 90)(57 85)(58 86)(59 87)(60 88)(67 125)(68 126)(69 121)(70 122)(71 123)(72 124)(91 118)(92 119)(93 120)(94 115)(95 116)(96 117)(103 185)(104 186)(105 181)(106 182)(107 183)(108 184)(133 146)(134 147)(135 148)(136 149)(137 150)(138 145)(199 208)(200 209)(201 210)(202 205)(203 206)(204 207)(211 237)(212 238)(213 239)(214 240)(215 235)(216 236)
(1 183 44 14 27)(2 184 45 15 28)(3 185 46 16 29)(4 186 47 17 30)(5 181 48 18 25)(6 182 43 13 26)(7 58 71 163 92)(8 59 72 164 93)(9 60 67 165 94)(10 55 68 166 95)(11 56 69 167 96)(12 57 70 168 91)(19 209 216 137 127)(20 210 211 138 128)(21 205 212 133 129)(22 206 213 134 130)(23 207 214 135 131)(24 208 215 136 132)(31 77 188 169 106)(32 78 189 170 107)(33 73 190 171 108)(34 74 191 172 103)(35 75 192 173 104)(36 76 187 174 105)(37 193 204 240 148)(38 194 199 235 149)(39 195 200 236 150)(40 196 201 237 145)(41 197 202 238 146)(42 198 203 239 147)(49 118 110 85 122)(50 119 111 86 123)(51 120 112 87 124)(52 115 113 88 125)(53 116 114 89 126)(54 117 109 90 121)(61 83 233 153 139)(62 84 234 154 140)(63 79 229 155 141)(64 80 230 156 142)(65 81 231 151 143)(66 82 232 152 144)(97 226 218 159 180)(98 227 219 160 175)(99 228 220 161 176)(100 223 221 162 177)(101 224 222 157 178)(102 225 217 158 179)
(1 195 111 19)(2 196 112 20)(3 197 113 21)(4 198 114 22)(5 193 109 23)(6 194 110 24)(7 141 170 220)(8 142 171 221)(9 143 172 222)(10 144 173 217)(11 139 174 218)(12 140 169 219)(13 235 49 215)(14 236 50 216)(15 237 51 211)(16 238 52 212)(17 239 53 213)(18 240 54 214)(25 204 117 207)(26 199 118 208)(27 200 119 209)(28 201 120 210)(29 202 115 205)(30 203 116 206)(31 98 70 234)(32 99 71 229)(33 100 72 230)(34 101 67 231)(35 102 68 232)(36 97 69 233)(37 90 131 181)(38 85 132 182)(39 86 127 183)(40 87 128 184)(41 88 129 185)(42 89 130 186)(43 149 122 136)(44 150 123 137)(45 145 124 138)(46 146 125 133)(47 147 126 134)(48 148 121 135)(55 152 104 225)(56 153 105 226)(57 154 106 227)(58 155 107 228)(59 156 108 223)(60 151 103 224)(61 187 159 96)(62 188 160 91)(63 189 161 92)(64 190 162 93)(65 191 157 94)(66 192 158 95)(73 177 164 80)(74 178 165 81)(75 179 166 82)(76 180 167 83)(77 175 168 84)(78 176 163 79)

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

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

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

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A 20B 20C 20D 24A ··· 24H 30A ··· 30L 30M ··· 30AB 60A ··· 60H order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 12 12 12 12 15 15 15 15 20 20 20 20 24 ··· 24 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 1 1 4 4 1 1 2 2 20 20 2 2 1 ··· 1 4 4 4 4 10 10 10 10 2 ··· 2 4 ··· 4 2 2 2 2 20 20 20 20 2 2 2 2 4 4 4 4 10 ··· 10 2 ··· 2 4 ··· 4 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 D4 D4 D5 SD16 D10 D10 C3×D4 C3×D4 C3×D5 C5⋊D4 C5⋊D4 C3×SD16 C6×D5 C6×D5 C3×C5⋊D4 C3×C5⋊D4 D4.D5 C3×D4.D5 kernel C6×D4.D5 C6×C5⋊2C8 C3×D4.D5 C6×Dic10 D4×C30 C2×D4.D5 C2×C5⋊2C8 D4.D5 C2×Dic10 D4×C10 C60 C2×C30 C6×D4 C30 C2×C12 C3×D4 C20 C2×C10 C2×D4 C12 C2×C6 C10 C2×C4 D4 C4 C22 C6 C2 # reps 1 1 4 1 1 2 2 8 2 2 1 1 2 4 2 4 2 2 4 4 4 8 4 8 8 8 4 8

Matrix representation of C6×D4.D5 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 225 0 0 0 0 0 0 225
,
 0 1 0 0 0 0 240 0 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 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 0 190 240 0 0 0 0 191 240
,
 19 222 0 0 0 0 222 222 0 0 0 0 0 0 0 240 0 0 0 0 240 0 0 0 0 0 0 0 125 17 0 0 0 0 130 116

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,225,0,0,0,0,0,0,225],[0,240,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,190,191,0,0,0,0,240,240],[19,222,0,0,0,0,222,222,0,0,0,0,0,0,0,240,0,0,0,0,240,0,0,0,0,0,0,0,125,130,0,0,0,0,17,116] >;

C6×D4.D5 in GAP, Magma, Sage, TeX

C_6\times D_4.D_5
% in TeX

G:=Group("C6xD4.D5");
// GroupNames label

G:=SmallGroup(480,726);
// by ID

G=gap.SmallGroup(480,726);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,336,590,2524,648,102,18822]);
// Polycyclic

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

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