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G = C12×D20order 480 = 25·3·5

Direct product of C12 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C12×D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C30 — D5×C2×C6 — C6×D20 — C12×D20
 Lower central C5 — C10 — C12×D20
 Upper central C1 — C2×C12 — C4×C12

Generators and relations for C12×D20
G = < a,b,c | a12=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 624 in 188 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], C5, C6 [×3], C6 [×4], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], C12 [×4], C12 [×3], C2×C6, C2×C6 [×8], C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×4], C20, D10 [×4], D10 [×4], C2×C10, C2×C12 [×3], C2×C12 [×6], C3×D4 [×4], C22×C6 [×2], C3×D5 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, C22×C12 [×2], C6×D4, C3×Dic5 [×2], C60 [×4], C60, C6×D5 [×4], C6×D5 [×4], C2×C30, C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C4×D5 [×2], C2×D20, D4×C12, D5×C12 [×4], C3×D20 [×4], C6×Dic5 [×2], C2×C60 [×3], D5×C2×C6 [×2], C4×D20, C3×C4⋊Dic5, C3×D10⋊C4 [×2], C4×C60, D5×C2×C12 [×2], C6×D20, C12×D20
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×2], C23, D5, C12 [×4], C2×C6 [×7], C22×C4, C2×D4, C4○D4, D10 [×3], C2×C12 [×6], C3×D4 [×2], C22×C6, C3×D5, C4×D4, C4×D5 [×2], D20 [×2], C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5 [×3], C2×C4×D5, C2×D20, C4○D20, D4×C12, D5×C12 [×2], C3×D20 [×2], D5×C2×C6, C4×D20, D5×C2×C12, C6×D20, C3×C4○D20, C12×D20

Smallest permutation representation of C12×D20
On 240 points
Generators in S240
(1 68 107 94 208 50 21 136 168 184 226 142)(2 69 108 95 209 51 22 137 169 185 227 143)(3 70 109 96 210 52 23 138 170 186 228 144)(4 71 110 97 211 53 24 139 171 187 229 145)(5 72 111 98 212 54 25 140 172 188 230 146)(6 73 112 99 213 55 26 121 173 189 231 147)(7 74 113 100 214 56 27 122 174 190 232 148)(8 75 114 81 215 57 28 123 175 191 233 149)(9 76 115 82 216 58 29 124 176 192 234 150)(10 77 116 83 217 59 30 125 177 193 235 151)(11 78 117 84 218 60 31 126 178 194 236 152)(12 79 118 85 219 41 32 127 179 195 237 153)(13 80 119 86 220 42 33 128 180 196 238 154)(14 61 120 87 201 43 34 129 161 197 239 155)(15 62 101 88 202 44 35 130 162 198 240 156)(16 63 102 89 203 45 36 131 163 199 221 157)(17 64 103 90 204 46 37 132 164 200 222 158)(18 65 104 91 205 47 38 133 165 181 223 159)(19 66 105 92 206 48 39 134 166 182 224 160)(20 67 106 93 207 49 40 135 167 183 225 141)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 35)(22 34)(23 33)(24 32)(25 31)(26 30)(27 29)(36 40)(37 39)(41 53)(42 52)(43 51)(44 50)(45 49)(46 48)(54 60)(55 59)(56 58)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 107)(102 106)(103 105)(108 120)(109 119)(110 118)(111 117)(112 116)(113 115)(121 125)(122 124)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(141 157)(142 156)(143 155)(144 154)(145 153)(146 152)(147 151)(148 150)(158 160)(161 169)(162 168)(163 167)(164 166)(170 180)(171 179)(172 178)(173 177)(174 176)(182 200)(183 199)(184 198)(185 197)(186 196)(187 195)(188 194)(189 193)(190 192)(201 209)(202 208)(203 207)(204 206)(210 220)(211 219)(212 218)(213 217)(214 216)(221 225)(222 224)(226 240)(227 239)(228 238)(229 237)(230 236)(231 235)(232 234)

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

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

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

156 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A ··· 6F 6G ··· 6N 10A ··· 10F 12A ··· 12H 12I ··· 12P 12Q ··· 12X 15A 15B 15C 15D 20A ··· 20X 30A ··· 30L 60A ··· 60AV order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 ··· 6 6 ··· 6 10 ··· 10 12 ··· 12 12 ··· 12 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 10 10 1 1 1 1 1 1 2 2 2 2 10 10 10 10 2 2 1 ··· 1 10 ··· 10 2 ··· 2 1 ··· 1 2 ··· 2 10 ··· 10 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

156 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 D4 D5 C4○D4 D10 C3×D4 C3×D5 C4×D5 D20 C3×C4○D4 C6×D5 C4○D20 D5×C12 C3×D20 C3×C4○D20 kernel C12×D20 C3×C4⋊Dic5 C3×D10⋊C4 C4×C60 D5×C2×C12 C6×D20 C4×D20 C3×D20 C4⋊Dic5 D10⋊C4 C4×C20 C2×C4×D5 C2×D20 D20 C60 C4×C12 C30 C2×C12 C20 C42 C12 C12 C10 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 2 8 2 4 2 4 2 16 2 2 2 6 4 4 8 8 4 12 8 16 16 16

Matrix representation of C12×D20 in GL4(𝔽61) generated by

 21 0 0 0 0 21 0 0 0 0 1 0 0 0 0 1
,
 18 60 0 0 1 0 0 0 0 0 29 36 0 0 27 2
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 45 60
G:=sub<GL(4,GF(61))| [21,0,0,0,0,21,0,0,0,0,1,0,0,0,0,1],[18,1,0,0,60,0,0,0,0,0,29,27,0,0,36,2],[0,1,0,0,1,0,0,0,0,0,1,45,0,0,0,60] >;

C12×D20 in GAP, Magma, Sage, TeX

C_{12}\times D_{20}
% in TeX

G:=Group("C12xD20");
// GroupNames label

G:=SmallGroup(480,666);
// by ID

G=gap.SmallGroup(480,666);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,142,18822]);
// Polycyclic

G:=Group<a,b,c|a^12=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽