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## G = C2×C60.C4order 480 = 25·3·5

### Direct product of C2 and C60.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×C60.C4
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C15⋊C8 — C2×C15⋊C8 — C2×C60.C4
 Lower central C15 — C2×C60.C4
 Upper central C1 — C2×C4

Generators and relations for C2×C60.C4
G = < a,b,c | a2=b60=1, c4=b30, ab=ba, ac=ca, cbc-1=b17 >

Subgroups: 524 in 152 conjugacy classes, 81 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C22×C8, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C22×C12, C3×Dic5, C60, C6×D5, C2×C30, D5⋊C8, C2×C5⋊C8, C2×C4×D5, C22×C3⋊C8, C15⋊C8, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, C2×D5⋊C8, C60.C4, C2×C15⋊C8, D5×C2×C12, C2×C60.C4
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, Dic3, D6, C2×C8, C22×C4, F5, C3⋊C8, C2×Dic3, C22×S3, C22×C8, C2×F5, C2×C3⋊C8, C22×Dic3, C3⋊F5, D5⋊C8, C22×F5, C22×C3⋊C8, C2×C3⋊F5, C2×D5⋊C8, C60.C4, C22×C3⋊F5, C2×C60.C4

Smallest permutation representation of C2×C60.C4
On 240 points
Generators in S240
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)(121 216)(122 217)(123 218)(124 219)(125 220)(126 221)(127 222)(128 223)(129 224)(130 225)(131 226)(132 227)(133 228)(134 229)(135 230)(136 231)(137 232)(138 233)(139 234)(140 235)(141 236)(142 237)(143 238)(144 239)(145 240)(146 181)(147 182)(148 183)(149 184)(150 185)(151 186)(152 187)(153 188)(154 189)(155 190)(156 191)(157 192)(158 193)(159 194)(160 195)(161 196)(162 197)(163 198)(164 199)(165 200)(166 201)(167 202)(168 203)(169 204)(170 205)(171 206)(172 207)(173 208)(174 209)(175 210)(176 211)(177 212)(178 213)(179 214)(180 215)
(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)
(1 167 117 187 31 137 87 217)(2 160 106 204 32 130 76 234)(3 153 95 221 33 123 65 191)(4 146 84 238 34 176 114 208)(5 139 73 195 35 169 103 225)(6 132 62 212 36 162 92 182)(7 125 111 229 37 155 81 199)(8 178 100 186 38 148 70 216)(9 171 89 203 39 141 119 233)(10 164 78 220 40 134 108 190)(11 157 67 237 41 127 97 207)(12 150 116 194 42 180 86 224)(13 143 105 211 43 173 75 181)(14 136 94 228 44 166 64 198)(15 129 83 185 45 159 113 215)(16 122 72 202 46 152 102 232)(17 175 61 219 47 145 91 189)(18 168 110 236 48 138 80 206)(19 161 99 193 49 131 69 223)(20 154 88 210 50 124 118 240)(21 147 77 227 51 177 107 197)(22 140 66 184 52 170 96 214)(23 133 115 201 53 163 85 231)(24 126 104 218 54 156 74 188)(25 179 93 235 55 149 63 205)(26 172 82 192 56 142 112 222)(27 165 71 209 57 135 101 239)(28 158 120 226 58 128 90 196)(29 151 109 183 59 121 79 213)(30 144 98 200 60 174 68 230)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - + + - - + + + image C1 C2 C2 C2 C4 C4 C4 C8 S3 Dic3 D6 D6 Dic3 Dic3 C3⋊C8 F5 C2×F5 C2×F5 C3⋊F5 D5⋊C8 C2×C3⋊F5 C2×C3⋊F5 C60.C4 kernel C2×C60.C4 C60.C4 C2×C15⋊C8 D5×C2×C12 D5×C12 C2×C60 D5×C2×C6 C6×D5 C2×C4×D5 C4×D5 C4×D5 C2×Dic5 C2×C20 C22×D5 D10 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 2 1 4 2 2 16 1 2 2 1 1 1 8 1 2 1 2 4 4 2 8

Matrix representation of C2×C60.C4 in GL6(𝔽241)

 240 0 0 0 0 0 0 240 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 177 177 0 0 0 0 64 0 0 0 0 0 0 0 196 175 66 45 0 0 196 130 0 111 0 0 0 130 196 45 0 0 66 175 196 0
,
 3 91 0 0 0 0 88 238 0 0 0 0 0 0 0 136 74 167 0 0 167 0 74 62 0 0 62 74 0 167 0 0 167 74 136 0

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[177,64,0,0,0,0,177,0,0,0,0,0,0,0,196,196,0,66,0,0,175,130,130,175,0,0,66,0,196,196,0,0,45,111,45,0],[3,88,0,0,0,0,91,238,0,0,0,0,0,0,0,167,62,167,0,0,136,0,74,74,0,0,74,74,0,136,0,0,167,62,167,0] >;

C2×C60.C4 in GAP, Magma, Sage, TeX

C_2\times C_{60}.C_4
% in TeX

G:=Group("C2xC60.C4");
// GroupNames label

G:=SmallGroup(480,1060);
// by ID

G=gap.SmallGroup(480,1060);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,100,80,2693,14118,2379]);
// Polycyclic

G:=Group<a,b,c|a^2=b^60=1,c^4=b^30,a*b=b*a,a*c=c*a,c*b*c^-1=b^17>;
// generators/relations

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