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## G = C4×D60order 480 = 25·3·5

### Direct product of C4 and D60

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C4×D60
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C22×D15 — C2×D60 — C4×D60
 Lower central C15 — C30 — C4×D60
 Upper central C1 — C2×C4 — C42

Generators and relations for C4×D60
G = < a,b,c | a4=b60=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 1188 in 188 conjugacy classes, 71 normal (39 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×3], C22, C22 [×8], C5, S3 [×4], C6 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], D5 [×4], C10 [×3], Dic3 [×2], C12 [×4], C12, D6 [×8], C2×C6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×4], C20, D10 [×8], C2×C10, C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×3], C22×S3 [×2], D15 [×4], C30 [×3], C4×D4, C4×D5 [×4], D20 [×4], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C4⋊Dic3, D6⋊C4 [×2], C4×C12, S3×C2×C4 [×2], C2×D12, Dic15 [×2], C60 [×4], C60, D30 [×4], D30 [×4], C2×C30, C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C4×D5 [×2], C2×D20, C4×D12, C4×D15 [×4], D60 [×4], C2×Dic15 [×2], C2×C60 [×3], C22×D15 [×2], C4×D20, C605C4, D303C4 [×2], C4×C60, C2×C4×D15 [×2], C2×D60, C4×D60
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D5, D6 [×3], C22×C4, C2×D4, C4○D4, D10 [×3], C4×S3 [×2], D12 [×2], C22×S3, D15, C4×D4, C4×D5 [×2], D20 [×2], C22×D5, S3×C2×C4, C2×D12, C4○D12, D30 [×3], C2×C4×D5, C2×D20, C4○D20, C4×D12, C4×D15 [×2], D60 [×2], C22×D15, C4×D20, C2×C4×D15, C2×D60, D6011C2, C4×D60

Smallest permutation representation of C4×D60
On 240 points
Generators in S240
(1 148 115 197)(2 149 116 198)(3 150 117 199)(4 151 118 200)(5 152 119 201)(6 153 120 202)(7 154 61 203)(8 155 62 204)(9 156 63 205)(10 157 64 206)(11 158 65 207)(12 159 66 208)(13 160 67 209)(14 161 68 210)(15 162 69 211)(16 163 70 212)(17 164 71 213)(18 165 72 214)(19 166 73 215)(20 167 74 216)(21 168 75 217)(22 169 76 218)(23 170 77 219)(24 171 78 220)(25 172 79 221)(26 173 80 222)(27 174 81 223)(28 175 82 224)(29 176 83 225)(30 177 84 226)(31 178 85 227)(32 179 86 228)(33 180 87 229)(34 121 88 230)(35 122 89 231)(36 123 90 232)(37 124 91 233)(38 125 92 234)(39 126 93 235)(40 127 94 236)(41 128 95 237)(42 129 96 238)(43 130 97 239)(44 131 98 240)(45 132 99 181)(46 133 100 182)(47 134 101 183)(48 135 102 184)(49 136 103 185)(50 137 104 186)(51 138 105 187)(52 139 106 188)(53 140 107 189)(54 141 108 190)(55 142 109 191)(56 143 110 192)(57 144 111 193)(58 145 112 194)(59 146 113 195)(60 147 114 196)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 120)(11 119)(12 118)(13 117)(14 116)(15 115)(16 114)(17 113)(18 112)(19 111)(20 110)(21 109)(22 108)(23 107)(24 106)(25 105)(26 104)(27 103)(28 102)(29 101)(30 100)(31 99)(32 98)(33 97)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 81)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(121 238)(122 237)(123 236)(124 235)(125 234)(126 233)(127 232)(128 231)(129 230)(130 229)(131 228)(132 227)(133 226)(134 225)(135 224)(136 223)(137 222)(138 221)(139 220)(140 219)(141 218)(142 217)(143 216)(144 215)(145 214)(146 213)(147 212)(148 211)(149 210)(150 209)(151 208)(152 207)(153 206)(154 205)(155 204)(156 203)(157 202)(158 201)(159 200)(160 199)(161 198)(162 197)(163 196)(164 195)(165 194)(166 193)(167 192)(168 191)(169 190)(170 189)(171 188)(172 187)(173 186)(174 185)(175 184)(176 183)(177 182)(178 181)(179 240)(180 239)

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

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

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

132 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 5A 5B 6A 6B 6C 10A ··· 10F 12A ··· 12L 15A 15B 15C 15D 20A ··· 20X 30A ··· 30L 60A ··· 60AV order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 30 30 30 30 2 1 1 1 1 2 2 2 2 30 30 30 30 2 2 2 2 2 2 ··· 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

132 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D5 D6 C4○D4 D10 C4×S3 D12 D15 C4×D5 D20 C4○D12 D30 C4○D20 C4×D15 D60 D60⋊11C2 kernel C4×D60 C60⋊5C4 D30⋊3C4 C4×C60 C2×C4×D15 C2×D60 D60 C4×C20 C60 C4×C12 C2×C20 C30 C2×C12 C20 C20 C42 C12 C12 C10 C2×C4 C6 C4 C4 C2 # reps 1 1 2 1 2 1 8 1 2 2 3 2 6 4 4 4 8 8 4 12 8 16 16 16

Matrix representation of C4×D60 in GL3(𝔽61) generated by

 50 0 0 0 11 0 0 0 11
,
 60 0 0 0 2 8 0 31 33
,
 1 0 0 0 53 28 0 13 8
G:=sub<GL(3,GF(61))| [50,0,0,0,11,0,0,0,11],[60,0,0,0,2,31,0,8,33],[1,0,0,0,53,13,0,28,8] >;

C4×D60 in GAP, Magma, Sage, TeX

C_4\times D_{60}
% in TeX

G:=Group("C4xD60");
// GroupNames label

G:=SmallGroup(480,838);
// by ID

G=gap.SmallGroup(480,838);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,58,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^4=b^60=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
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