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G = D64Dic10order 480 = 25·3·5

4th semidirect product of D6 and Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D64Dic10, Dic15.9D4, D6⋊C4.9D5, C4⋊Dic57S3, (S3×C10)⋊4Q8, C6.42(D4×D5), C54(D6⋊Q8), (C2×C20).26D6, C10.43(S3×D4), C30.54(C2×D4), C10.36(S3×Q8), C30.50(C2×Q8), C1521(C22⋊Q8), (C2×C12).228D10, D6⋊Dic5.14C2, C30.4Q816C2, Dic155C422C2, (C2×Dic5).42D6, C2.18(S3×Dic10), C6.18(C2×Dic10), C30.123(C4○D4), C10.72(C4○D12), C6.47(D42D5), C2.19(C20⋊D6), (C2×C30).126C23, (C2×C60).259C22, (C2×Dic3).41D10, (C22×S3).44D10, C33(Dic5.14D4), (C6×Dic5).78C22, C2.18(Dic3.D10), (C10×Dic3).79C22, (C2×Dic15).100C22, (C2×C15⋊Q8)⋊11C2, (C5×D6⋊C4).9C2, (C2×C4).57(S3×D5), (C2×S3×Dic5).8C2, (C3×C4⋊Dic5)⋊18C2, C22.187(C2×S3×D5), (S3×C2×C10).25C22, (C2×C6).138(C22×D5), (C2×C10).138(C22×S3), SmallGroup(480,512)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D64Dic10
C1C5C15C30C2×C30C6×Dic5C2×S3×Dic5 — D64Dic10
C15C2×C30 — D64Dic10
C1C22C2×C4

Generators and relations for D64Dic10
 G = < a,b,c,d | a6=b2=c20=1, d2=c10, bab=cac-1=dad-1=a-1, cbc-1=ab, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 716 in 148 conjugacy classes, 50 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×4], C12 [×3], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×5], C20 [×2], C2×C10, C2×C10 [×4], Dic6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C22×C10, Dic3⋊C4 [×2], D6⋊C4, D6⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], Dic15, C60, S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, C23.D5, C5×C22⋊C4, C2×Dic10, C22×Dic5, D6⋊Q8, S3×Dic5 [×2], C15⋊Q8 [×2], C6×Dic5 [×2], C10×Dic3, C2×Dic15 [×2], C2×C60, S3×C2×C10, Dic5.14D4, D6⋊Dic5, Dic155C4, C3×C4⋊Dic5, C5×D6⋊C4, C30.4Q8, C2×S3×Dic5, C2×C15⋊Q8, D64Dic10
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], C22×S3, C22⋊Q8, Dic10 [×2], C22×D5, C4○D12, S3×D4, S3×Q8, S3×D5, C2×Dic10, D4×D5, D42D5, D6⋊Q8, C2×S3×D5, Dic5.14D4, S3×Dic10, C20⋊D6, Dic3.D10, D64Dic10

Smallest permutation representation of D64Dic10
On 240 points
Generators in S240
(1 95 196 205 138 39)(2 40 139 206 197 96)(3 97 198 207 140 21)(4 22 121 208 199 98)(5 99 200 209 122 23)(6 24 123 210 181 100)(7 81 182 211 124 25)(8 26 125 212 183 82)(9 83 184 213 126 27)(10 28 127 214 185 84)(11 85 186 215 128 29)(12 30 129 216 187 86)(13 87 188 217 130 31)(14 32 131 218 189 88)(15 89 190 219 132 33)(16 34 133 220 191 90)(17 91 192 201 134 35)(18 36 135 202 193 92)(19 93 194 203 136 37)(20 38 137 204 195 94)(41 180 229 73 109 153)(42 154 110 74 230 161)(43 162 231 75 111 155)(44 156 112 76 232 163)(45 164 233 77 113 157)(46 158 114 78 234 165)(47 166 235 79 115 159)(48 160 116 80 236 167)(49 168 237 61 117 141)(50 142 118 62 238 169)(51 170 239 63 119 143)(52 144 120 64 240 171)(53 172 221 65 101 145)(54 146 102 66 222 173)(55 174 223 67 103 147)(56 148 104 68 224 175)(57 176 225 69 105 149)(58 150 106 70 226 177)(59 178 227 71 107 151)(60 152 108 72 228 179)
(1 29)(2 129)(3 31)(4 131)(5 33)(6 133)(7 35)(8 135)(9 37)(10 137)(11 39)(12 139)(13 21)(14 121)(15 23)(16 123)(17 25)(18 125)(19 27)(20 127)(22 32)(24 34)(26 36)(28 38)(30 40)(41 170)(42 52)(43 172)(44 54)(45 174)(46 56)(47 176)(48 58)(49 178)(50 60)(51 180)(53 162)(55 164)(57 166)(59 168)(61 107)(62 72)(63 109)(64 74)(65 111)(66 76)(67 113)(68 78)(69 115)(70 80)(71 117)(73 119)(75 101)(77 103)(79 105)(81 134)(82 202)(83 136)(84 204)(85 138)(86 206)(87 140)(88 208)(89 122)(90 210)(91 124)(92 212)(93 126)(94 214)(95 128)(96 216)(97 130)(98 218)(99 132)(100 220)(102 232)(104 234)(106 236)(108 238)(110 240)(112 222)(114 224)(116 226)(118 228)(120 230)(141 227)(142 179)(143 229)(144 161)(145 231)(146 163)(147 233)(148 165)(149 235)(150 167)(151 237)(152 169)(153 239)(154 171)(155 221)(156 173)(157 223)(158 175)(159 225)(160 177)(181 191)(182 201)(183 193)(184 203)(185 195)(186 205)(187 197)(188 207)(189 199)(190 209)(192 211)(194 213)(196 215)(198 217)(200 219)
(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)
(1 164 11 174)(2 163 12 173)(3 162 13 172)(4 161 14 171)(5 180 15 170)(6 179 16 169)(7 178 17 168)(8 177 18 167)(9 176 19 166)(10 175 20 165)(21 231 31 221)(22 230 32 240)(23 229 33 239)(24 228 34 238)(25 227 35 237)(26 226 36 236)(27 225 37 235)(28 224 38 234)(29 223 39 233)(30 222 40 232)(41 89 51 99)(42 88 52 98)(43 87 53 97)(44 86 54 96)(45 85 55 95)(46 84 56 94)(47 83 57 93)(48 82 58 92)(49 81 59 91)(50 100 60 90)(61 124 71 134)(62 123 72 133)(63 122 73 132)(64 121 74 131)(65 140 75 130)(66 139 76 129)(67 138 77 128)(68 137 78 127)(69 136 79 126)(70 135 80 125)(101 207 111 217)(102 206 112 216)(103 205 113 215)(104 204 114 214)(105 203 115 213)(106 202 116 212)(107 201 117 211)(108 220 118 210)(109 219 119 209)(110 218 120 208)(141 182 151 192)(142 181 152 191)(143 200 153 190)(144 199 154 189)(145 198 155 188)(146 197 156 187)(147 196 157 186)(148 195 158 185)(149 194 159 184)(150 193 160 183)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G10H10I10J12A12B12C12D12E12F15A15B20A20B20C20D20E20F20G20H30A···30F60A···60H
order1222223444444445566610···10101010101212121212121515202020202020202030···3060···60
size1111662410101220303060222222···2121212124420202020444444121212124···44···4

60 irreducible representations

dim11111111222222222222444444444
type++++++++++-++++++-+-++-+-
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10D10Dic10C4○D12S3×D4S3×Q8S3×D5D4×D5D42D5C2×S3×D5S3×Dic10C20⋊D6Dic3.D10
kernelD64Dic10D6⋊Dic5Dic155C4C3×C4⋊Dic5C5×D6⋊C4C30.4Q8C2×S3×Dic5C2×C15⋊Q8C4⋊Dic5Dic15S3×C10D6⋊C4C2×Dic5C2×C20C30C2×Dic3C2×C12C22×S3D6C10C10C10C2×C4C6C6C22C2C2C2
# reps11111111122221222284112222444

Matrix representation of D64Dic10 in GL6(𝔽61)

100000
010000
0060000
0006000
00005941
0000521
,
6000000
0600000
001000
0036000
00005941
0000522
,
54290000
32590000
00163000
00264500
000010
0000960
,
56510000
2750000
0050000
00281100
0000600
0000521

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,59,52,0,0,0,0,41,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,3,0,0,0,0,0,60,0,0,0,0,0,0,59,52,0,0,0,0,41,2],[54,32,0,0,0,0,29,59,0,0,0,0,0,0,16,26,0,0,0,0,30,45,0,0,0,0,0,0,1,9,0,0,0,0,0,60],[56,27,0,0,0,0,51,5,0,0,0,0,0,0,50,28,0,0,0,0,0,11,0,0,0,0,0,0,60,52,0,0,0,0,0,1] >;

D64Dic10 in GAP, Magma, Sage, TeX

D_6\rtimes_4{\rm Dic}_{10}
% in TeX

G:=Group("D6:4Dic10");
// GroupNames label

G:=SmallGroup(480,512);
// by ID

G=gap.SmallGroup(480,512);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,590,219,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^20=1,d^2=c^10,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a*b,d*b*d^-1=a^4*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽