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## G = C6×D40order 480 = 25·3·5

### Direct product of C6 and D40

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C6×D40
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C6×D20 — C6×D40
 Lower central C5 — C10 — C20 — C6×D40
 Upper central C1 — C2×C6 — C2×C12 — C2×C24

Generators and relations for C6×D40
G = < a,b,c | a6=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 752 in 152 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, C6, C6 [×2], C6 [×4], C8 [×2], C2×C4, D4 [×6], C23 [×2], D5 [×4], C10, C10 [×2], C12 [×2], C2×C6, C2×C6 [×8], C15, C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], D10 [×8], C2×C10, C24 [×2], C2×C12, C3×D4 [×6], C22×C6 [×2], C3×D5 [×4], C30, C30 [×2], C2×D8, C40 [×2], D20 [×4], D20 [×2], C2×C20, C22×D5 [×2], C2×C24, C3×D8 [×4], C6×D4 [×2], C60 [×2], C6×D5 [×8], C2×C30, D40 [×4], C2×C40, C2×D20 [×2], C6×D8, C120 [×2], C3×D20 [×4], C3×D20 [×2], C2×C60, D5×C2×C6 [×2], C2×D40, C3×D40 [×4], C2×C120, C6×D20 [×2], C6×D40
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], C23, D5, C2×C6 [×7], D8 [×2], C2×D4, D10 [×3], C3×D4 [×2], C22×C6, C3×D5, C2×D8, D20 [×2], C22×D5, C3×D8 [×2], C6×D4, C6×D5 [×3], D40 [×2], C2×D20, C6×D8, C3×D20 [×2], D5×C2×C6, C2×D40, C3×D40 [×2], C6×D20, C6×D40

Smallest permutation representation of C6×D40
On 240 points
Generators in S240
(1 66 178 99 229 135)(2 67 179 100 230 136)(3 68 180 101 231 137)(4 69 181 102 232 138)(5 70 182 103 233 139)(6 71 183 104 234 140)(7 72 184 105 235 141)(8 73 185 106 236 142)(9 74 186 107 237 143)(10 75 187 108 238 144)(11 76 188 109 239 145)(12 77 189 110 240 146)(13 78 190 111 201 147)(14 79 191 112 202 148)(15 80 192 113 203 149)(16 41 193 114 204 150)(17 42 194 115 205 151)(18 43 195 116 206 152)(19 44 196 117 207 153)(20 45 197 118 208 154)(21 46 198 119 209 155)(22 47 199 120 210 156)(23 48 200 81 211 157)(24 49 161 82 212 158)(25 50 162 83 213 159)(26 51 163 84 214 160)(27 52 164 85 215 121)(28 53 165 86 216 122)(29 54 166 87 217 123)(30 55 167 88 218 124)(31 56 168 89 219 125)(32 57 169 90 220 126)(33 58 170 91 221 127)(34 59 171 92 222 128)(35 60 172 93 223 129)(36 61 173 94 224 130)(37 62 174 95 225 131)(38 63 175 96 226 132)(39 64 176 97 227 133)(40 65 177 98 228 134)
(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 98)(2 97)(3 96)(4 95)(5 94)(6 93)(7 92)(8 91)(9 90)(10 89)(11 88)(12 87)(13 86)(14 85)(15 84)(16 83)(17 82)(18 81)(19 120)(20 119)(21 118)(22 117)(23 116)(24 115)(25 114)(26 113)(27 112)(28 111)(29 110)(30 109)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 102)(38 101)(39 100)(40 99)(41 213)(42 212)(43 211)(44 210)(45 209)(46 208)(47 207)(48 206)(49 205)(50 204)(51 203)(52 202)(53 201)(54 240)(55 239)(56 238)(57 237)(58 236)(59 235)(60 234)(61 233)(62 232)(63 231)(64 230)(65 229)(66 228)(67 227)(68 226)(69 225)(70 224)(71 223)(72 222)(73 221)(74 220)(75 219)(76 218)(77 217)(78 216)(79 215)(80 214)(121 191)(122 190)(123 189)(124 188)(125 187)(126 186)(127 185)(128 184)(129 183)(130 182)(131 181)(132 180)(133 179)(134 178)(135 177)(136 176)(137 175)(138 174)(139 173)(140 172)(141 171)(142 170)(143 169)(144 168)(145 167)(146 166)(147 165)(148 164)(149 163)(150 162)(151 161)(152 200)(153 199)(154 198)(155 197)(156 196)(157 195)(158 194)(159 193)(160 192)

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

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

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

138 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 6A ··· 6F 6G ··· 6N 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 6 ··· 6 6 ··· 6 8 8 8 8 10 ··· 10 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 20 20 1 1 2 2 2 2 1 ··· 1 20 ··· 20 2 2 2 2 2 ··· 2 2 2 2 2 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 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D4 D5 D8 D10 D10 C3×D4 C3×D4 C3×D5 D20 D20 C3×D8 C6×D5 C6×D5 D40 C3×D20 C3×D20 C3×D40 kernel C6×D40 C3×D40 C2×C120 C6×D20 C2×D40 D40 C2×C40 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 2 2 8 2 4 1 1 2 4 4 2 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C6×D40 in GL5(𝔽241)

 240 0 0 0 0 0 15 0 0 0 0 0 15 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 219 55 0 0 0 92 0 0 0 0 0 0 199 227 0 0 0 14 194
,
 1 0 0 0 0 0 219 55 0 0 0 92 22 0 0 0 0 0 221 29 0 0 0 194 20

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,15,0,0,0,0,0,15,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,219,92,0,0,0,55,0,0,0,0,0,0,199,14,0,0,0,227,194],[1,0,0,0,0,0,219,92,0,0,0,55,22,0,0,0,0,0,221,194,0,0,0,29,20] >;

C6×D40 in GAP, Magma, Sage, TeX

C_6\times D_{40}
% in TeX

G:=Group("C6xD40");
// GroupNames label

G:=SmallGroup(480,696);
// by ID

G=gap.SmallGroup(480,696);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,590,394,2524,102,18822]);
// 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^-1>;
// generators/relations

׿
×
𝔽