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G = Dic12⋊D5order 480 = 25·3·5

1st semidirect product of Dic12 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C30.9D8, C40.9D6, C156SD32, C10.8D24, C60.51D4, C20.3D12, Dic121D5, D120.4C2, C24.42D10, C120.19C22, C52C163S3, C8.18(S3×D5), C53(C48⋊C2), C6.3(D4⋊D5), C31(C5⋊SD32), (C5×Dic12)⋊1C2, C2.6(C5⋊D24), C4.3(C5⋊D12), C12.53(C5⋊D4), (C3×C52C16)⋊3C2, SmallGroup(480,21)

Series: Derived Chief Lower central Upper central

C1C120 — Dic12⋊D5
C1C5C15C30C60C120C3×C52C16 — Dic12⋊D5
C15C30C60C120 — Dic12⋊D5
C1C2C4C8

Generators and relations for Dic12⋊D5
 G = < a,b,c,d | a24=c5=d2=1, b2=a12, bab-1=dad=a-1, ac=ca, bc=cb, dbd=a21b, dcd=c-1 >

120C2
12C4
60C22
40S3
24D5
6Q8
30D4
4Dic3
20D6
12D10
12C20
8D15
3Q16
5C16
15D8
2Dic6
10D12
6C5×Q8
6D20
4D30
4C5×Dic3
15SD32
5C48
5D24
3C5×Q16
3D40
2D60
2C5×Dic6
5C48⋊C2
3C5⋊SD32

Smallest permutation representation of Dic12⋊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)
(1 137 13 125)(2 136 14 124)(3 135 15 123)(4 134 16 122)(5 133 17 121)(6 132 18 144)(7 131 19 143)(8 130 20 142)(9 129 21 141)(10 128 22 140)(11 127 23 139)(12 126 24 138)(25 203 37 215)(26 202 38 214)(27 201 39 213)(28 200 40 212)(29 199 41 211)(30 198 42 210)(31 197 43 209)(32 196 44 208)(33 195 45 207)(34 194 46 206)(35 193 47 205)(36 216 48 204)(49 183 61 171)(50 182 62 170)(51 181 63 169)(52 180 64 192)(53 179 65 191)(54 178 66 190)(55 177 67 189)(56 176 68 188)(57 175 69 187)(58 174 70 186)(59 173 71 185)(60 172 72 184)(73 105 85 117)(74 104 86 116)(75 103 87 115)(76 102 88 114)(77 101 89 113)(78 100 90 112)(79 99 91 111)(80 98 92 110)(81 97 93 109)(82 120 94 108)(83 119 95 107)(84 118 96 106)(145 223 157 235)(146 222 158 234)(147 221 159 233)(148 220 160 232)(149 219 161 231)(150 218 162 230)(151 217 163 229)(152 240 164 228)(153 239 165 227)(154 238 166 226)(155 237 167 225)(156 236 168 224)
(1 199 99 150 70)(2 200 100 151 71)(3 201 101 152 72)(4 202 102 153 49)(5 203 103 154 50)(6 204 104 155 51)(7 205 105 156 52)(8 206 106 157 53)(9 207 107 158 54)(10 208 108 159 55)(11 209 109 160 56)(12 210 110 161 57)(13 211 111 162 58)(14 212 112 163 59)(15 213 113 164 60)(16 214 114 165 61)(17 215 115 166 62)(18 216 116 167 63)(19 193 117 168 64)(20 194 118 145 65)(21 195 119 146 66)(22 196 120 147 67)(23 197 97 148 68)(24 198 98 149 69)(25 75 226 170 121)(26 76 227 171 122)(27 77 228 172 123)(28 78 229 173 124)(29 79 230 174 125)(30 80 231 175 126)(31 81 232 176 127)(32 82 233 177 128)(33 83 234 178 129)(34 84 235 179 130)(35 85 236 180 131)(36 86 237 181 132)(37 87 238 182 133)(38 88 239 183 134)(39 89 240 184 135)(40 90 217 185 136)(41 91 218 186 137)(42 92 219 187 138)(43 93 220 188 139)(44 94 221 189 140)(45 95 222 190 141)(46 96 223 191 142)(47 73 224 192 143)(48 74 225 169 144)
(1 70)(2 69)(3 68)(4 67)(5 66)(6 65)(7 64)(8 63)(9 62)(10 61)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 50)(22 49)(23 72)(24 71)(25 237)(26 236)(27 235)(28 234)(29 233)(30 232)(31 231)(32 230)(33 229)(34 228)(35 227)(36 226)(37 225)(38 224)(39 223)(40 222)(41 221)(42 220)(43 219)(44 218)(45 217)(46 240)(47 239)(48 238)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(89 96)(90 95)(91 94)(92 93)(97 101)(98 100)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(121 181)(122 180)(123 179)(124 178)(125 177)(126 176)(127 175)(128 174)(129 173)(130 172)(131 171)(132 170)(133 169)(134 192)(135 191)(136 190)(137 189)(138 188)(139 187)(140 186)(141 185)(142 184)(143 183)(144 182)(145 204)(146 203)(147 202)(148 201)(149 200)(150 199)(151 198)(152 197)(153 196)(154 195)(155 194)(156 193)(157 216)(158 215)(159 214)(160 213)(161 212)(162 211)(163 210)(164 209)(165 208)(166 207)(167 206)(168 205)

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

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

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

57 conjugacy classes

class 1 2A2B 3 4A4B5A5B 6 8A8B10A10B12A12B15A15B16A16B16C16D20A20B20C20D20E20F24A24B24C24D30A30B40A40B40C40D48A···48H60A60B60C60D120A···120H
order12234455688101012121515161616162020202020202424242430304040404048···4860606060120···120
size11120222422222222244101010104424242424222244444410···1044444···4

57 irreducible representations

dim111122222222222444444
type++++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10D12SD32C5⋊D4D24C48⋊C2S3×D5D4⋊D5C5⋊D12C5⋊SD32C5⋊D24Dic12⋊D5
kernelDic12⋊D5C3×C52C16C5×Dic12D120C52C16C60Dic12C40C30C24C20C15C12C10C5C8C6C4C3C2C1
# reps111111212224448222448

Matrix representation of Dic12⋊D5 in GL4(𝔽241) generated by

14521300
15721900
002400
000240
,
2304400
521100
00165103
008976
,
1000
0100
002401
0050190
,
240000
192100
002400
00501
G:=sub<GL(4,GF(241))| [145,157,0,0,213,219,0,0,0,0,240,0,0,0,0,240],[230,52,0,0,44,11,0,0,0,0,165,89,0,0,103,76],[1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[240,192,0,0,0,1,0,0,0,0,240,50,0,0,0,1] >;

Dic12⋊D5 in GAP, Magma, Sage, TeX

{\rm Dic}_{12}\rtimes D_5
% in TeX

G:=Group("Dic12:D5");
// GroupNames label

G:=SmallGroup(480,21);
// by ID

G=gap.SmallGroup(480,21);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,309,120,135,142,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^24=c^5=d^2=1,b^2=a^12,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a^21*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of Dic12⋊D5 in TeX

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