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## G = C3×C40.C4order 480 = 25·3·5

### Direct product of C3 and C40.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C40.C4
 Chief series C1 — C5 — C10 — C20 — C4×D5 — D5×C12 — C3×C4.F5 — C3×C40.C4
 Lower central C5 — C10 — C20 — C3×C40.C4
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×C40.C4
G = < a,b,c | a3=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 184 in 60 conjugacy classes, 32 normal (28 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), Dic5, C20, D10, C24, C24, C2×C12, C3×D5, C30, C8.C4, C52C8, C40, C5⋊C8, C4×D5, C2×C24, C3×M4(2), C3×Dic5, C60, C6×D5, C8×D5, C4.F5, C3×C8.C4, C3×C52C8, C120, C3×C5⋊C8, D5×C12, C40.C4, D5×C24, C3×C4.F5, C3×C40.C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C4⋊C4, F5, C2×C12, C3×D4, C3×Q8, C8.C4, C2×F5, C3×C4⋊C4, C3×F5, C4⋊F5, C3×C8.C4, C6×F5, C40.C4, C3×C4⋊F5, C3×C40.C4

Smallest permutation representation of C3×C40.C4
On 240 points
Generators in S240
(1 134 227)(2 135 228)(3 136 229)(4 137 230)(5 138 231)(6 139 232)(7 140 233)(8 141 234)(9 142 235)(10 143 236)(11 144 237)(12 145 238)(13 146 239)(14 147 240)(15 148 201)(16 149 202)(17 150 203)(18 151 204)(19 152 205)(20 153 206)(21 154 207)(22 155 208)(23 156 209)(24 157 210)(25 158 211)(26 159 212)(27 160 213)(28 121 214)(29 122 215)(30 123 216)(31 124 217)(32 125 218)(33 126 219)(34 127 220)(35 128 221)(36 129 222)(37 130 223)(38 131 224)(39 132 225)(40 133 226)(41 173 112)(42 174 113)(43 175 114)(44 176 115)(45 177 116)(46 178 117)(47 179 118)(48 180 119)(49 181 120)(50 182 81)(51 183 82)(52 184 83)(53 185 84)(54 186 85)(55 187 86)(56 188 87)(57 189 88)(58 190 89)(59 191 90)(60 192 91)(61 193 92)(62 194 93)(63 195 94)(64 196 95)(65 197 96)(66 198 97)(67 199 98)(68 200 99)(69 161 100)(70 162 101)(71 163 102)(72 164 103)(73 165 104)(74 166 105)(75 167 106)(76 168 107)(77 169 108)(78 170 109)(79 171 110)(80 172 111)
(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 62 31 72 21 42 11 52)(2 49 40 75 22 69 20 55)(3 76 9 78 23 56 29 58)(4 63 18 41 24 43 38 61)(5 50 27 44 25 70 7 64)(6 77 36 47 26 57 16 67)(8 51 14 53 28 71 34 73)(10 65 32 59 30 45 12 79)(13 66 19 68 33 46 39 48)(15 80 37 74 35 60 17 54)(81 213 115 211 101 233 95 231)(82 240 84 214 102 220 104 234)(83 227 93 217 103 207 113 237)(85 201 111 223 105 221 91 203)(86 228 120 226 106 208 100 206)(87 215 89 229 107 235 109 209)(88 202 98 232 108 222 118 212)(90 216 116 238 110 236 96 218)(92 230 94 204 112 210 114 224)(97 205 99 219 117 225 119 239)(121 163 127 165 141 183 147 185)(122 190 136 168 142 170 156 188)(123 177 145 171 143 197 125 191)(124 164 154 174 144 184 134 194)(126 178 132 180 146 198 152 200)(128 192 150 186 148 172 130 166)(129 179 159 189 149 199 139 169)(131 193 137 195 151 173 157 175)(133 167 155 161 153 187 135 181)(138 182 160 176 158 162 140 196)

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

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

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

66 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 8G 8H 10 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 24A 24B 24C 24D 24E 24F 24G 24H 24I ··· 24P 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 3 3 4 4 4 5 6 6 6 6 8 8 8 8 8 8 8 8 10 12 12 12 12 12 12 15 15 20 20 24 24 24 24 24 24 24 24 24 ··· 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 10 1 1 2 5 5 4 1 1 10 10 2 2 10 10 20 20 20 20 4 2 2 5 5 5 5 4 4 4 4 2 2 2 2 10 10 10 10 20 ··· 20 4 4 4 4 4 4 4 4 4 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + - + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 D4 Q8 C3×D4 C3×Q8 C8.C4 C3×C8.C4 F5 C2×F5 C3×F5 C4⋊F5 C6×F5 C40.C4 C3×C4⋊F5 C3×C40.C4 kernel C3×C40.C4 D5×C24 C3×C4.F5 C40.C4 C3×C5⋊2C8 C120 C8×D5 C4.F5 C5⋊2C8 C40 C3×Dic5 C6×D5 Dic5 D10 C15 C5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 2 2 4 4 4 1 1 2 2 4 8 1 1 2 2 2 4 4 8

Matrix representation of C3×C40.C4 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15 0 0 0 0 0 0 15
,
 30 0 0 0 0 0 0 8 0 0 0 0 0 0 207 207 0 224 0 0 17 224 224 17 0 0 224 0 207 207 0 0 34 17 34 0
,
 0 1 0 0 0 0 64 0 0 0 0 0 0 0 0 108 195 108 0 0 87 0 133 133 0 0 133 133 0 87 0 0 108 195 108 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15,0,0,0,0,0,0,15],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,207,17,224,34,0,0,207,224,0,17,0,0,0,224,207,34,0,0,224,17,207,0],[0,64,0,0,0,0,1,0,0,0,0,0,0,0,0,87,133,108,0,0,108,0,133,195,0,0,195,133,0,108,0,0,108,133,87,0] >;

C3×C40.C4 in GAP, Magma, Sage, TeX

C_3\times C_{40}.C_4
% in TeX

G:=Group("C3xC40.C4");
// GroupNames label

G:=SmallGroup(480,275);
// by ID

G=gap.SmallGroup(480,275);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,136,2524,102,9414,1595]);
// Polycyclic

G:=Group<a,b,c|a^3=b^40=1,c^4=b^20,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

׿
×
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