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## G = C6×C40⋊C2order 480 = 25·3·5

### Direct product of C6 and C40⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C6×C40⋊C2
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C6×D20 — C6×C40⋊C2
 Lower central C5 — C10 — C20 — C6×C40⋊C2
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C6×C40⋊C2
G = < a,b,c | a6=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

Subgroups: 560 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C24 [×2], C2×C12, C2×C12, C3×D4 [×3], C3×Q8 [×3], C22×C6, C3×D5 [×2], C30, C30 [×2], C2×SD16, C40 [×2], Dic10 [×2], Dic10, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C2×C24, C3×SD16 [×4], C6×D4, C6×Q8, C3×Dic5 [×2], C60 [×2], C6×D5 [×4], C2×C30, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C6×SD16, C120 [×2], C3×Dic10 [×2], C3×Dic10, C3×D20 [×2], C3×D20, C6×Dic5, C2×C60, D5×C2×C6, C2×C40⋊C2, C3×C40⋊C2 [×4], C2×C120, C6×Dic10, C6×D20, C6×C40⋊C2
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], SD16 [×2], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C2×SD16, D20 [×2], C22×D5, C3×SD16 [×2], C6×D4, C6×D5 [×3], C40⋊C2 [×2], C2×D20, C6×SD16, C3×D20 [×2], D5×C2×C6, C2×C40⋊C2, C3×C40⋊C2 [×2], C6×D20, C6×C40⋊C2

Smallest permutation representation of C6×C40⋊C2
On 240 points
Generators in S240
(1 77 189 209 101 144)(2 78 190 210 102 145)(3 79 191 211 103 146)(4 80 192 212 104 147)(5 41 193 213 105 148)(6 42 194 214 106 149)(7 43 195 215 107 150)(8 44 196 216 108 151)(9 45 197 217 109 152)(10 46 198 218 110 153)(11 47 199 219 111 154)(12 48 200 220 112 155)(13 49 161 221 113 156)(14 50 162 222 114 157)(15 51 163 223 115 158)(16 52 164 224 116 159)(17 53 165 225 117 160)(18 54 166 226 118 121)(19 55 167 227 119 122)(20 56 168 228 120 123)(21 57 169 229 81 124)(22 58 170 230 82 125)(23 59 171 231 83 126)(24 60 172 232 84 127)(25 61 173 233 85 128)(26 62 174 234 86 129)(27 63 175 235 87 130)(28 64 176 236 88 131)(29 65 177 237 89 132)(30 66 178 238 90 133)(31 67 179 239 91 134)(32 68 180 240 92 135)(33 69 181 201 93 136)(34 70 182 202 94 137)(35 71 183 203 95 138)(36 72 184 204 96 139)(37 73 185 205 97 140)(38 74 186 206 98 141)(39 75 187 207 99 142)(40 76 188 208 100 143)
(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)
(1 209)(2 228)(3 207)(4 226)(5 205)(6 224)(7 203)(8 222)(9 201)(10 220)(11 239)(12 218)(13 237)(14 216)(15 235)(16 214)(17 233)(18 212)(19 231)(20 210)(21 229)(22 208)(23 227)(24 206)(25 225)(26 204)(27 223)(28 202)(29 221)(30 240)(31 219)(32 238)(33 217)(34 236)(35 215)(36 234)(37 213)(38 232)(39 211)(40 230)(41 97)(42 116)(43 95)(44 114)(45 93)(46 112)(47 91)(48 110)(49 89)(50 108)(51 87)(52 106)(53 85)(54 104)(55 83)(56 102)(57 81)(58 100)(59 119)(60 98)(61 117)(62 96)(63 115)(64 94)(65 113)(66 92)(67 111)(68 90)(69 109)(70 88)(71 107)(72 86)(73 105)(74 84)(75 103)(76 82)(77 101)(78 120)(79 99)(80 118)(121 192)(122 171)(123 190)(124 169)(125 188)(126 167)(127 186)(128 165)(129 184)(130 163)(131 182)(132 161)(133 180)(134 199)(135 178)(136 197)(137 176)(138 195)(139 174)(140 193)(141 172)(142 191)(143 170)(144 189)(145 168)(146 187)(147 166)(148 185)(149 164)(150 183)(151 162)(152 181)(153 200)(154 179)(155 198)(156 177)(157 196)(158 175)(159 194)(160 173)

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

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

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

138 conjugacy classes

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

138 irreducible representations

 dim 1 1 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 C2 C2 C3 C6 C6 C6 C6 D4 D4 D5 SD16 D10 D10 C3×D4 C3×D4 C3×D5 D20 D20 C3×SD16 C6×D5 C6×D5 C40⋊C2 C3×D20 C3×D20 C3×C40⋊C2 kernel C6×C40⋊C2 C3×C40⋊C2 C2×C120 C6×Dic10 C6×D20 C2×C40⋊C2 C40⋊C2 C2×C40 C2×Dic10 C2×D20 C60 C2×C30 C2×C24 C30 C24 C2×C12 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 1 1 2 8 2 2 2 1 1 2 4 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C6×C40⋊C2 in GL4(𝔽241) generated by

 226 0 0 0 0 226 0 0 0 0 16 0 0 0 0 16
,
 0 189 0 0 51 189 0 0 0 0 204 68 0 0 173 125
,
 52 188 0 0 51 189 0 0 0 0 240 52 0 0 0 1
G:=sub<GL(4,GF(241))| [226,0,0,0,0,226,0,0,0,0,16,0,0,0,0,16],[0,51,0,0,189,189,0,0,0,0,204,173,0,0,68,125],[52,51,0,0,188,189,0,0,0,0,240,0,0,0,52,1] >;

C6×C40⋊C2 in GAP, Magma, Sage, TeX

C_6\times C_{40}\rtimes C_2
% in TeX

G:=Group("C6xC40:C2");
// GroupNames label

G:=SmallGroup(480,695);
// by ID

G=gap.SmallGroup(480,695);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,142,2524,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^6=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations

׿
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