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G = C4×C60order 240 = 24·3·5

Abelian group of type [4,60]

Aliases: C4×C60, SmallGroup(240,81)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4×C60
 Chief series C1 — C2 — C22 — C2×C10 — C2×C30 — C2×C60 — C4×C60
 Lower central C1 — C4×C60
 Upper central C1 — C4×C60

Generators and relations for C4×C60
G = < a,b | a4=b60=1, ab=ba >

Subgroups: 60, all normal (12 characteristic)
C1, C2 [×3], C3, C4 [×6], C22, C5, C6 [×3], C2×C4 [×3], C10 [×3], C12 [×6], C2×C6, C15, C42, C20 [×6], C2×C10, C2×C12 [×3], C30 [×3], C2×C20 [×3], C4×C12, C60 [×6], C2×C30, C4×C20, C2×C60 [×3], C4×C60
Quotients: C1, C2 [×3], C3, C4 [×6], C22, C5, C6 [×3], C2×C4 [×3], C10 [×3], C12 [×6], C2×C6, C15, C42, C20 [×6], C2×C10, C2×C12 [×3], C30 [×3], C2×C20 [×3], C4×C12, C60 [×6], C2×C30, C4×C20, C2×C60 [×3], C4×C60

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

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

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

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

C4×C60 is a maximal subgroup of
C42.D15  C605C8  D607C4  C608Q8  C60.24Q8  C422D15  C426D15  C427D15  C423D15

240 conjugacy classes

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

240 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C5 C6 C10 C12 C15 C20 C30 C60 kernel C4×C60 C2×C60 C4×C20 C60 C4×C12 C2×C20 C2×C12 C20 C42 C12 C2×C4 C4 # reps 1 3 2 12 4 6 12 24 8 48 24 96

Matrix representation of C4×C60 in GL2(𝔽61) generated by

 60 0 0 11
,
 2 0 0 21
G:=sub<GL(2,GF(61))| [60,0,0,11],[2,0,0,21] >;

C4×C60 in GAP, Magma, Sage, TeX

C_4\times C_{60}
% in TeX

G:=Group("C4xC60");
// GroupNames label

G:=SmallGroup(240,81);
// by ID

G=gap.SmallGroup(240,81);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,360,727]);
// Polycyclic

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

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