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## G = (C2×C60).C4order 480 = 25·3·5

### 2nd non-split extension by C2×C60 of C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — (C2×C60).C4
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — C6×Dic5 — C15⋊8M4(2) — (C2×C60).C4
 Lower central C15 — C30 — C2×C30 — (C2×C60).C4
 Upper central C1 — C2 — C22 — C2×C4

Generators and relations for (C2×C60).C4
G = < a,b,c | a2=b60=1, c4=b30, ab=ba, cac-1=ab30, cbc-1=ab47 >

Subgroups: 332 in 76 conjugacy classes, 29 normal (23 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C2×C6, C15, M4(2), C2×Q8, Dic5, Dic5, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C5⋊C8, Dic10, C2×Dic5, C2×C20, C4.Dic3, C6×Q8, C3×Dic5, C3×Dic5, C60, C2×C30, C22.F5, C2×Dic10, C12.10D4, C15⋊C8, C3×Dic10, C6×Dic5, C2×C60, Dic5.D4, C158M4(2), C6×Dic10, (C2×C60).C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, F5, C2×Dic3, C3⋊D4, C4.10D4, C2×F5, C6.D4, C3⋊F5, C22⋊F5, C12.10D4, C2×C3⋊F5, Dic5.D4, D10.D6, (C2×C60).C4

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + - + - + - + + - image C1 C2 C2 C4 C4 S3 D4 Dic3 D6 Dic3 C3⋊D4 F5 C4.10D4 C2×F5 C3⋊F5 C22⋊F5 C12.10D4 C2×C3⋊F5 Dic5.D4 D10.D6 (C2×C60).C4 kernel (C2×C60).C4 C15⋊8M4(2) C6×Dic10 C6×Dic5 C2×C60 C2×Dic10 C3×Dic5 C2×Dic5 C2×Dic5 C2×C20 Dic5 C2×C12 C15 C2×C6 C2×C4 C6 C5 C22 C3 C2 C1 # reps 1 2 1 2 2 1 2 1 1 1 4 1 1 1 2 2 2 2 4 4 8

Matrix representation of (C2×C60).C4 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 240 0 0 0 0 240
,
 45 98 0 0 80 206 0 0 0 0 43 174 0 0 24 19
,
 0 0 1 0 0 0 0 1 210 68 0 0 78 31 0 0
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,0,0,0,0,240],[45,80,0,0,98,206,0,0,0,0,43,24,0,0,174,19],[0,0,210,78,0,0,68,31,1,0,0,0,0,1,0,0] >;

(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(480,310);
// by ID

G=gap.SmallGroup(480,310);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,675,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c|a^2=b^60=1,c^4=b^30,a*b=b*a,c*a*c^-1=a*b^30,c*b*c^-1=a*b^47>;
// generators/relations

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