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## G = C3×C40.6C4order 480 = 25·3·5

### Direct product of C3 and C40.6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×C40.6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C3×C4.Dic5 — C3×C40.6C4
 Lower central C5 — C10 — C20 — C3×C40.6C4
 Upper central C1 — C12 — C2×C12 — C2×C24

Generators and relations for C3×C40.6C4
G = < a,b,c | a3=b40=1, c4=b20, ab=ba, ac=ca, cbc-1=b19 >

Subgroups: 128 in 60 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C8 [×2], C8 [×2], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C24 [×2], C24 [×2], C2×C12, C30, C30, C8.C4, C52C8 [×2], C40 [×2], C2×C20, C2×C24, C3×M4(2) [×2], C60 [×2], C2×C30, C4.Dic5 [×2], C2×C40, C3×C8.C4, C3×C52C8 [×2], C120 [×2], C2×C60, C40.6C4, C3×C4.Dic5 [×2], C2×C120, C3×C40.6C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C6 [×3], C2×C4, D4, Q8, D5, C12 [×2], C2×C6, C4⋊C4, Dic5 [×2], D10, C2×C12, C3×D4, C3×Q8, C3×D5, C8.C4, Dic10, D20, C2×Dic5, C3×C4⋊C4, C3×Dic5 [×2], C6×D5, C4⋊Dic5, C3×C8.C4, C3×Dic10, C3×D20, C6×Dic5, C40.6C4, C3×C4⋊Dic5, C3×C40.6C4

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

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

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

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

138 conjugacy classes

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

138 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + + - image C1 C2 C2 C3 C4 C6 C6 C12 D4 Q8 D5 Dic5 D10 C3×D4 C3×Q8 C3×D5 C8.C4 D20 Dic10 C3×Dic5 C6×D5 C3×C8.C4 C3×D20 C3×Dic10 C40.6C4 C3×C40.6C4 kernel C3×C40.6C4 C3×C4.Dic5 C2×C120 C40.6C4 C120 C4.Dic5 C2×C40 C40 C60 C2×C30 C2×C24 C24 C2×C12 C20 C2×C10 C2×C8 C15 C12 C2×C6 C8 C2×C4 C5 C4 C22 C3 C1 # reps 1 2 1 2 4 4 2 8 1 1 2 4 2 2 2 4 4 4 4 8 4 8 8 8 16 32

Matrix representation of C3×C40.6C4 in GL2(𝔽241) generated by

 225 0 0 225
,
 193 0 0 236
,
 0 1 64 0
G:=sub<GL(2,GF(241))| [225,0,0,225],[193,0,0,236],[0,64,1,0] >;

C3×C40.6C4 in GAP, Magma, Sage, TeX

C_3\times C_{40}._6C_4
% in TeX

G:=Group("C3xC40.6C4");
// GroupNames label

G:=SmallGroup(480,97);
// by ID

G=gap.SmallGroup(480,97);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,84,365,176,136,2524,102,18822]);
// 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^19>;
// generators/relations

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