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G = D101Dic6order 480 = 25·3·5

1st semidirect product of D10 and Dic6 acting via Dic6/Dic3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D101Dic6, (C6×D5)⋊1Q8, C6.33(Q8×D5), (C2×Dic6)⋊4D5, C30.42(C2×Q8), C33(D103Q8), C30.134(C2×D4), C10.130(S3×D4), (C2×C20).226D6, (C2×C12).18D10, C1514(C22⋊Q8), (C5×Dic3).6D4, C2.15(D5×Dic6), (C10×Dic6)⋊14C2, D10⋊C4.8S3, C30.64(C4○D4), C6.Dic1017C2, C30.Q823C2, C30.4Q818C2, (C2×Dic5).34D6, C10.15(C2×Dic6), (C22×D5).49D6, (C2×C30).111C23, (C2×C60).319C22, C6.35(Q82D5), C56(Dic3.D4), Dic3.8(C5⋊D4), C10.13(D42S3), C2.17(D20⋊S3), (C2×Dic3).104D10, D10⋊Dic3.11C2, (C6×Dic5).65C22, (C2×Dic15).89C22, (C10×Dic3).69C22, (C2×C4).47(S3×D5), C6.31(C2×C5⋊D4), C2.12(S3×C5⋊D4), (C2×D5×Dic3).5C2, (D5×C2×C6).19C22, C22.177(C2×S3×D5), (C2×C6).123(C22×D5), (C3×D10⋊C4).11C2, (C2×C10).123(C22×S3), SmallGroup(480,497)

Series: Derived Chief Lower central Upper central

C1C2×C30 — D101Dic6
C1C5C15C30C2×C30D5×C2×C6C2×D5×Dic3 — D101Dic6
C15C2×C30 — D101Dic6
C1C22C2×C4

Generators and relations for D101Dic6
 G = < a,b,c,d | a10=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, cbc-1=a5b, bd=db, dcd-1=c-1 >

Subgroups: 684 in 148 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×4], D10 [×2], D10 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×4], C2×C12, C2×C12, C22×C6, C3×D5 [×2], C30 [×3], C22⋊Q8, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, C5×Dic3 [×2], C5×Dic3, C3×Dic5, Dic15 [×2], C60, C6×D5 [×2], C6×D5 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, D10⋊C4, D10⋊C4, C2×C4×D5, Q8×C10, Dic3.D4, D5×Dic3 [×2], C6×Dic5, C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, D103Q8, D10⋊Dic3, C30.Q8, C6.Dic10, C3×D10⋊C4, C30.4Q8, C2×D5×Dic3, C10×Dic6, D101Dic6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C22×S3, C22⋊Q8, C5⋊D4 [×2], C22×D5, C2×Dic6, S3×D4, D42S3, S3×D5, Q8×D5, Q82D5, C2×C5⋊D4, Dic3.D4, C2×S3×D5, D103Q8, D5×Dic6, D20⋊S3, S3×C5⋊D4, D101Dic6

Smallest permutation representation of D101Dic6
On 240 points
Generators in S240
(1 181 108 81 18 159 58 156 180 194)(2 182 97 82 19 160 59 145 169 195)(3 183 98 83 20 161 60 146 170 196)(4 184 99 84 21 162 49 147 171 197)(5 185 100 73 22 163 50 148 172 198)(6 186 101 74 23 164 51 149 173 199)(7 187 102 75 24 165 52 150 174 200)(8 188 103 76 13 166 53 151 175 201)(9 189 104 77 14 167 54 152 176 202)(10 190 105 78 15 168 55 153 177 203)(11 191 106 79 16 157 56 154 178 204)(12 192 107 80 17 158 57 155 179 193)(25 42 229 71 212 130 109 141 222 86)(26 43 230 72 213 131 110 142 223 87)(27 44 231 61 214 132 111 143 224 88)(28 45 232 62 215 121 112 144 225 89)(29 46 233 63 216 122 113 133 226 90)(30 47 234 64 205 123 114 134 227 91)(31 48 235 65 206 124 115 135 228 92)(32 37 236 66 207 125 116 136 217 93)(33 38 237 67 208 126 117 137 218 94)(34 39 238 68 209 127 118 138 219 95)(35 40 239 69 210 128 119 139 220 96)(36 41 240 70 211 129 120 140 221 85)
(1 200)(2 13)(3 202)(4 15)(5 204)(6 17)(7 194)(8 19)(9 196)(10 21)(11 198)(12 23)(14 161)(16 163)(18 165)(20 167)(22 157)(24 159)(25 31)(26 125)(27 33)(28 127)(29 35)(30 129)(32 131)(34 121)(36 123)(37 213)(38 88)(39 215)(40 90)(41 205)(42 92)(43 207)(44 94)(45 209)(46 96)(47 211)(48 86)(49 177)(50 79)(51 179)(52 81)(53 169)(54 83)(55 171)(56 73)(57 173)(58 75)(59 175)(60 77)(61 137)(62 238)(63 139)(64 240)(65 141)(66 230)(67 143)(68 232)(69 133)(70 234)(71 135)(72 236)(74 192)(76 182)(78 184)(80 186)(82 188)(84 190)(85 114)(87 116)(89 118)(91 120)(93 110)(95 112)(97 103)(98 152)(99 105)(100 154)(101 107)(102 156)(104 146)(106 148)(108 150)(109 206)(111 208)(113 210)(115 212)(117 214)(119 216)(122 128)(124 130)(126 132)(134 221)(136 223)(138 225)(140 227)(142 217)(144 219)(145 151)(147 153)(149 155)(158 199)(160 201)(162 203)(164 193)(166 195)(168 197)(170 189)(172 191)(174 181)(176 183)(178 185)(180 187)(218 231)(220 233)(222 235)(224 237)(226 239)(228 229)
(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)
(1 227 7 221)(2 226 8 220)(3 225 9 219)(4 224 10 218)(5 223 11 217)(6 222 12 228)(13 239 19 233)(14 238 20 232)(15 237 21 231)(16 236 22 230)(17 235 23 229)(18 234 24 240)(25 107 31 101)(26 106 32 100)(27 105 33 99)(28 104 34 98)(29 103 35 97)(30 102 36 108)(37 73 43 79)(38 84 44 78)(39 83 45 77)(40 82 46 76)(41 81 47 75)(42 80 48 74)(49 214 55 208)(50 213 56 207)(51 212 57 206)(52 211 58 205)(53 210 59 216)(54 209 60 215)(61 168 67 162)(62 167 68 161)(63 166 69 160)(64 165 70 159)(65 164 71 158)(66 163 72 157)(85 181 91 187)(86 192 92 186)(87 191 93 185)(88 190 94 184)(89 189 95 183)(90 188 96 182)(109 179 115 173)(110 178 116 172)(111 177 117 171)(112 176 118 170)(113 175 119 169)(114 174 120 180)(121 152 127 146)(122 151 128 145)(123 150 129 156)(124 149 130 155)(125 148 131 154)(126 147 132 153)(133 201 139 195)(134 200 140 194)(135 199 141 193)(136 198 142 204)(137 197 143 203)(138 196 144 202)

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E10A···10F12A12B12C12D15A15B20A20B20C20D20E···20L30A···30F60A···60H
order122222344444444556666610···101212121215152020202020···2030···3060···60
size11111010246612203030602222220202···244202044444412···124···44···4

60 irreducible representations

dim11111111222222222222444444444
type++++++++++-++++++-+-+-++-
imageC1C2C2C2C2C2C2C2S3D4Q8D5D6D6D6C4○D4D10D10Dic6C5⋊D4S3×D4D42S3S3×D5Q8×D5Q82D5C2×S3×D5D5×Dic6D20⋊S3S3×C5⋊D4
kernelD101Dic6D10⋊Dic3C30.Q8C6.Dic10C3×D10⋊C4C30.4Q8C2×D5×Dic3C10×Dic6D10⋊C4C5×Dic3C6×D5C2×Dic6C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12D10Dic3C10C10C2×C4C6C6C22C2C2C2
# reps11111111122211124248112222444

Matrix representation of D101Dic6 in GL6(𝔽61)

17170000
4410000
001000
000100
0000600
0000060
,
17170000
1440000
001000
000100
000010
00001660
,
47160000
45140000
0006000
001100
0000215
00005840
,
6000000
0600000
0044800
00251700
0000110
00005450

G:=sub<GL(6,GF(61))| [17,44,0,0,0,0,17,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[17,1,0,0,0,0,17,44,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,60],[47,45,0,0,0,0,16,14,0,0,0,0,0,0,0,1,0,0,0,0,60,1,0,0,0,0,0,0,21,58,0,0,0,0,5,40],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,25,0,0,0,0,8,17,0,0,0,0,0,0,11,54,0,0,0,0,0,50] >;

D101Dic6 in GAP, Magma, Sage, TeX

D_{10}\rtimes_1{\rm Dic}_6
% in TeX

G:=Group("D10:1Dic6");
// GroupNames label

G:=SmallGroup(480,497);
// by ID

G=gap.SmallGroup(480,497);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽