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## G = C2×C60.7C4order 480 = 25·3·5

### Direct product of C2 and C60.7C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×C60.7C4
 Chief series C1 — C5 — C15 — C30 — C60 — C15⋊3C8 — C2×C15⋊3C8 — C2×C60.7C4
 Lower central C15 — C30 — C2×C60.7C4
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C60.7C4
G = < a,b,c | a2=b60=1, c4=b30, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 372 in 136 conjugacy classes, 87 normal (37 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C8, C2×C4, C2×C4, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C2×C8, M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C3⋊C8, C2×C12, C2×C12, C22×C6, C30, C30, C30, C2×M4(2), C52C8, C2×C20, C2×C20, C22×C10, C2×C3⋊C8, C4.Dic3, C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C2×C52C8, C4.Dic5, C22×C20, C2×C4.Dic3, C153C8, C2×C60, C2×C60, C22×C30, C2×C4.Dic5, C2×C153C8, C60.7C4, C22×C60, C2×C60.7C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, M4(2), C22×C4, Dic5, D10, C2×Dic3, C22×S3, D15, C2×M4(2), C2×Dic5, C22×D5, C4.Dic3, C22×Dic3, Dic15, D30, C4.Dic5, C22×Dic5, C2×C4.Dic3, C2×Dic15, C22×D15, C2×C4.Dic5, C60.7C4, C22×Dic15, C2×C60.7C4

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

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

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

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

132 conjugacy classes

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

132 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - - + - + - + - image C1 C2 C2 C2 C4 C4 S3 D5 Dic3 D6 Dic3 M4(2) Dic5 D10 Dic5 D15 C4.Dic3 Dic15 D30 Dic15 C4.Dic5 C60.7C4 kernel C2×C60.7C4 C2×C15⋊3C8 C60.7C4 C22×C60 C2×C60 C22×C30 C22×C20 C22×C12 C2×C20 C2×C20 C22×C10 C30 C2×C12 C2×C12 C22×C6 C22×C4 C10 C2×C4 C2×C4 C23 C6 C2 # reps 1 2 4 1 6 2 1 2 3 3 1 4 6 6 2 4 8 12 12 4 16 32

Matrix representation of C2×C60.7C4 in GL3(𝔽241) generated by

 240 0 0 0 240 0 0 0 240
,
 240 0 0 0 90 0 0 147 158
,
 1 0 0 0 135 210 0 236 106
G:=sub<GL(3,GF(241))| [240,0,0,0,240,0,0,0,240],[240,0,0,0,90,147,0,0,158],[1,0,0,0,135,236,0,210,106] >;

C2×C60.7C4 in GAP, Magma, Sage, TeX

C_2\times C_{60}._7C_4
% in TeX

G:=Group("C2xC60.7C4");
// GroupNames label

G:=SmallGroup(480,886);
// by ID

G=gap.SmallGroup(480,886);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,80,2693,18822]);
// 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^-1>;
// generators/relations

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