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G = C3×C20.4C8order 480 = 25·3·5

Direct product of C3 and C20.4C8

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C3×C20.4C8, C40.9C12, C60.10C8, C20.4C24, C120.19C4, C24.83D10, C1514M5(2), C24.6Dic5, C120.101C22, C52C165C6, (C2×C30).9C8, C8.22(C6×D5), C54(C3×M5(2)), (C2×C60).43C4, (C2×C10).5C24, (C2×C40).10C6, C30.66(C2×C8), C40.22(C2×C6), (C2×C24).17D5, C8.2(C3×Dic5), C12.4(C52C8), C60.249(C2×C4), C20.61(C2×C12), (C2×C120).26C2, C10.18(C2×C24), (C2×C20).20C12, C4.11(C6×Dic5), (C2×C12).14Dic5, C12.50(C2×Dic5), C4.(C3×C52C8), C2.4(C6×C52C8), (C2×C8).7(C3×D5), C22.(C3×C52C8), C6.14(C2×C52C8), (C3×C52C16)⋊12C2, (C2×C6).1(C52C8), (C2×C4).5(C3×Dic5), SmallGroup(480,90)

Series: Derived Chief Lower central Upper central

C1C10 — C3×C20.4C8
C1C5C10C20C40C120C3×C52C16 — C3×C20.4C8
C5C10 — C3×C20.4C8
C1C24C2×C24

Generators and relations for C3×C20.4C8
 G = < a,b,c | a3=b40=1, c4=b10, ab=ba, ac=ca, cbc-1=b29 >

2C2
2C6
2C10
2C30
5C16
5C16
5M5(2)
5C48
5C48
5C3×M5(2)

Smallest permutation representation of C3×C20.4C8
On 240 points
Generators in S240
(1 112 66)(2 113 67)(3 114 68)(4 115 69)(5 116 70)(6 117 71)(7 118 72)(8 119 73)(9 120 74)(10 81 75)(11 82 76)(12 83 77)(13 84 78)(14 85 79)(15 86 80)(16 87 41)(17 88 42)(18 89 43)(19 90 44)(20 91 45)(21 92 46)(22 93 47)(23 94 48)(24 95 49)(25 96 50)(26 97 51)(27 98 52)(28 99 53)(29 100 54)(30 101 55)(31 102 56)(32 103 57)(33 104 58)(34 105 59)(35 106 60)(36 107 61)(37 108 62)(38 109 63)(39 110 64)(40 111 65)(121 221 180)(122 222 181)(123 223 182)(124 224 183)(125 225 184)(126 226 185)(127 227 186)(128 228 187)(129 229 188)(130 230 189)(131 231 190)(132 232 191)(133 233 192)(134 234 193)(135 235 194)(136 236 195)(137 237 196)(138 238 197)(139 239 198)(140 240 199)(141 201 200)(142 202 161)(143 203 162)(144 204 163)(145 205 164)(146 206 165)(147 207 166)(148 208 167)(149 209 168)(150 210 169)(151 211 170)(152 212 171)(153 213 172)(154 214 173)(155 215 174)(156 216 175)(157 217 176)(158 218 177)(159 219 178)(160 220 179)
(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)
(1 144 26 149 11 154 36 159 21 124 6 129 31 134 16 139)(2 133 27 138 12 143 37 148 22 153 7 158 32 123 17 128)(3 122 28 127 13 132 38 137 23 142 8 147 33 152 18 157)(4 151 29 156 14 121 39 126 24 131 9 136 34 141 19 146)(5 140 30 145 15 150 40 155 25 160 10 125 35 130 20 135)(41 198 66 163 51 168 76 173 61 178 46 183 71 188 56 193)(42 187 67 192 52 197 77 162 62 167 47 172 72 177 57 182)(43 176 68 181 53 186 78 191 63 196 48 161 73 166 58 171)(44 165 69 170 54 175 79 180 64 185 49 190 74 195 59 200)(45 194 70 199 55 164 80 169 65 174 50 179 75 184 60 189)(81 225 106 230 91 235 116 240 101 205 86 210 111 215 96 220)(82 214 107 219 92 224 117 229 102 234 87 239 112 204 97 209)(83 203 108 208 93 213 118 218 103 223 88 228 113 233 98 238)(84 232 109 237 94 202 119 207 104 212 89 217 114 222 99 227)(85 221 110 226 95 231 120 236 105 201 90 206 115 211 100 216)

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

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

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

156 conjugacy classes

class 1 2A2B3A3B4A4B4C5A5B6A6B6C6D8A8B8C8D8E8F10A···10F12A12B12C12D12E12F15A15B15C15D16A···16H20A···20H24A···24H24I24J24K24L30A···30L40A···40P48A···48P60A···60P120A···120AF
order1223344455666688888810···101212121212121515151516···1620···2024···242424242430···3040···4048···4860···60120···120
size112111122211221111222···2111122222210···102···21···122222···22···210···102···22···2

156 irreducible representations

dim111111111111112222222222222222
type++++-+-
imageC1C2C2C3C4C4C6C6C8C8C12C12C24C24D5Dic5D10Dic5C3×D5M5(2)C52C8C52C8C3×Dic5C6×D5C3×Dic5C3×M5(2)C3×C52C8C3×C52C8C20.4C8C3×C20.4C8
kernelC3×C20.4C8C3×C52C16C2×C120C20.4C8C120C2×C60C52C16C2×C40C60C2×C30C40C2×C20C20C2×C10C2×C24C24C24C2×C12C2×C8C15C12C2×C6C8C8C2×C4C5C4C22C3C1
# reps12122242444488222244444448881632

Matrix representation of C3×C20.4C8 in GL2(𝔽241) generated by

150
015
,
1160
124200
,
6490
238177
G:=sub<GL(2,GF(241))| [15,0,0,15],[116,124,0,200],[64,238,90,177] >;

C3×C20.4C8 in GAP, Magma, Sage, TeX

C_3\times C_{20}._4C_8
% in TeX

G:=Group("C3xC20.4C8");
// GroupNames label

G:=SmallGroup(480,90);
// by ID

G=gap.SmallGroup(480,90);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,701,80,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^3=b^40=1,c^4=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^29>;
// generators/relations

Export

Subgroup lattice of C3×C20.4C8 in TeX

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