Copied to
clipboard

G = C127D20order 480 = 25·3·5

1st semidirect product of C12 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C608D4, C127D20, D103D12, (C6×D5)⋊11D4, C4⋊Dic39D5, C6.23(D4×D5), (C2×D60)⋊26C2, C42(C3⋊D20), C157(C4⋊D4), C33(C4⋊D20), C51(C127D4), C202(C3⋊D4), C2.25(D5×D12), C30.60(C2×D4), C6.61(C2×D20), C10.24(C2×D12), (C2×C20).130D6, D304C418C2, C30.85(C4○D4), (C2×C12).306D10, (C22×D5).92D6, C10.15(C4○D12), (C2×C60).150C22, (C2×C30).140C23, C6.17(Q82D5), (C2×Dic5).184D6, (C2×Dic3).44D10, C2.17(C12.28D10), (C10×Dic3).87C22, (C6×Dic5).211C22, (C22×D15).47C22, (C2×C4×D5)⋊2S3, (D5×C2×C12)⋊3C2, (C2×C3⋊D20)⋊3C2, (C5×C4⋊Dic3)⋊6C2, (C2×C4).163(S3×D5), C2.19(C2×C3⋊D20), C10.16(C2×C3⋊D4), C22.192(C2×S3×D5), (D5×C2×C6).108C22, (C2×C6).152(C22×D5), (C2×C10).152(C22×S3), SmallGroup(480,526)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C127D20
C1C5C15C30C2×C30D5×C2×C6C2×C3⋊D20 — C127D20
C15C2×C30 — C127D20
C1C22C2×C4

Generators and relations for C127D20
 G = < a,b,c | a12=b20=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 1276 in 188 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×3], C22, C22 [×10], C5, S3 [×2], C6 [×3], C6 [×2], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], D5 [×4], C10 [×3], Dic3 [×2], C12 [×2], C12, D6 [×6], C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], Dic5, C20 [×2], C20 [×2], D10 [×2], D10 [×8], C2×C10, D12 [×2], C2×Dic3 [×2], C3⋊D4 [×4], C2×C12, C2×C12 [×3], C22×S3 [×2], C22×C6, C3×D5 [×2], D15 [×2], C30 [×3], C4⋊D4, C4×D5 [×2], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C22×D5, C22×D5 [×2], C4⋊Dic3, D6⋊C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5, C60 [×2], C6×D5 [×2], C6×D5 [×2], D30 [×6], C2×C30, D10⋊C4 [×2], C5×C4⋊C4, C2×C4×D5, C2×D20 [×3], C127D4, C3⋊D20 [×4], D5×C12 [×2], C6×Dic5, C10×Dic3 [×2], D60 [×2], C2×C60, D5×C2×C6, C22×D15 [×2], C4⋊D20, D304C4 [×2], C5×C4⋊Dic3, C2×C3⋊D20 [×2], D5×C2×C12, C2×D60, C127D20
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D5, D6 [×3], C2×D4 [×2], C4○D4, D10 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4, D20 [×2], C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, Q82D5, C127D4, C3⋊D20 [×2], C2×S3×D5, C4⋊D20, C12.28D10, D5×D12, C2×C3⋊D20, C127D20

Smallest permutation representation of C127D20
On 240 points
Generators in S240
(1 232 172 94 33 193 47 132 150 69 108 212)(2 213 109 70 151 133 48 194 34 95 173 233)(3 234 174 96 35 195 49 134 152 71 110 214)(4 215 111 72 153 135 50 196 36 97 175 235)(5 236 176 98 37 197 51 136 154 73 112 216)(6 217 113 74 155 137 52 198 38 99 177 237)(7 238 178 100 39 199 53 138 156 75 114 218)(8 219 115 76 157 139 54 200 40 81 179 239)(9 240 180 82 21 181 55 140 158 77 116 220)(10 201 117 78 159 121 56 182 22 83 161 221)(11 222 162 84 23 183 57 122 160 79 118 202)(12 203 119 80 141 123 58 184 24 85 163 223)(13 224 164 86 25 185 59 124 142 61 120 204)(14 205 101 62 143 125 60 186 26 87 165 225)(15 226 166 88 27 187 41 126 144 63 102 206)(16 207 103 64 145 127 42 188 28 89 167 227)(17 228 168 90 29 189 43 128 146 65 104 208)(18 209 105 66 147 129 44 190 30 91 169 229)(19 230 170 92 31 191 45 130 148 67 106 210)(20 211 107 68 149 131 46 192 32 93 171 231)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)(53 60)(54 59)(55 58)(56 57)(61 76)(62 75)(63 74)(64 73)(65 72)(66 71)(67 70)(68 69)(77 80)(78 79)(81 86)(82 85)(83 84)(87 100)(88 99)(89 98)(90 97)(91 96)(92 95)(93 94)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 122)(123 140)(124 139)(125 138)(126 137)(127 136)(128 135)(129 134)(130 133)(131 132)(141 158)(142 157)(143 156)(144 155)(145 154)(146 153)(147 152)(148 151)(149 150)(159 160)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 184)(182 183)(185 200)(186 199)(187 198)(188 197)(189 196)(190 195)(191 194)(192 193)(201 202)(203 220)(204 219)(205 218)(206 217)(207 216)(208 215)(209 214)(210 213)(211 212)(221 222)(223 240)(224 239)(225 238)(226 237)(227 236)(228 235)(229 234)(230 233)(231 232)

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F5A5B6A6B6C6D6E6F6G10A···10F12A12B12C12D12E12F12G12H15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222222344444455666666610···10121212121212121215152020202020···2030···3060···60
size1111101060602221010121222222101010102···222221010101044444412···124···44···4

66 irreducible representations

dim111111222222222222224444444
type++++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D6C4○D4D10D10C3⋊D4D12D20C4○D12S3×D5D4×D5Q82D5C3⋊D20C2×S3×D5C12.28D10D5×D12
kernelC127D20D304C4C5×C4⋊Dic3C2×C3⋊D20D5×C2×C12C2×D60C2×C4×D5C60C6×D5C4⋊Dic3C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C20D10C12C10C2×C4C6C6C4C22C2C2
# reps121211122211124244842224244

Matrix representation of C127D20 in GL4(𝔽61) generated by

1000
0100
004022
00029
,
73200
29200
00315
001530
,
73200
295400
00303
004631
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,22,29],[7,29,0,0,32,2,0,0,0,0,31,15,0,0,5,30],[7,29,0,0,32,54,0,0,0,0,30,46,0,0,3,31] >;

C127D20 in GAP, Magma, Sage, TeX

C_{12}\rtimes_7D_{20}
% in TeX

G:=Group("C12:7D20");
// GroupNames label

G:=SmallGroup(480,526);
// by ID

G=gap.SmallGroup(480,526);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,100,1356,18822]);
// Polycyclic

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

׿
×
𝔽