Copied to
clipboard

## G = C60.45D4order 480 = 25·3·5

### 45th non-split extension by C60 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C30 — C60.45D4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — D6⋊Dic5 — C60.45D4
 Lower central C15 — C2×C30 — C60.45D4
 Upper central C1 — C22 — C2×C4

Generators and relations for C60.45D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=a19, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 652 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, S3 [×2], C6 [×3], C2×C4, C2×C4 [×7], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×3], C12 [×2], C12 [×2], D6 [×2], D6 [×2], C2×C6, C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×4], Dic6 [×2], C4×S3 [×2], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3, C5×S3 [×2], C30 [×3], C22⋊Q8, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×3], C22×C10, C4⋊Dic3 [×2], D6⋊C4 [×2], C3×C4⋊C4, C2×Dic6, S3×C2×C4, C5×Dic3, C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×2], S3×C10 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, C4.D12, C6×Dic5 [×2], S3×C20 [×2], C10×Dic3, Dic30 [×2], C2×Dic15 [×2], C2×C60, S3×C2×C10, C20.48D4, D6⋊Dic5 [×2], C30.Q8 [×2], C3×C4⋊Dic5, S3×C2×C20, C2×Dic30, C60.45D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], D12 [×2], C22×S3, C22⋊Q8, Dic10 [×2], C5⋊D4 [×2], C22×D5, C2×D12, D42S3, S3×Q8, S3×D5, C2×Dic10, C4○D20, C2×C5⋊D4, C4.D12, C5⋊D12 [×2], C2×S3×D5, C20.48D4, D205S3, S3×Dic10, C2×C5⋊D12, C60.45D4

Smallest permutation representation of C60.45D4
On 240 points
Generators in S240
(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 91 132 192)(2 110 133 211)(3 69 134 230)(4 88 135 189)(5 107 136 208)(6 66 137 227)(7 85 138 186)(8 104 139 205)(9 63 140 224)(10 82 141 183)(11 101 142 202)(12 120 143 221)(13 79 144 240)(14 98 145 199)(15 117 146 218)(16 76 147 237)(17 95 148 196)(18 114 149 215)(19 73 150 234)(20 92 151 193)(21 111 152 212)(22 70 153 231)(23 89 154 190)(24 108 155 209)(25 67 156 228)(26 86 157 187)(27 105 158 206)(28 64 159 225)(29 83 160 184)(30 102 161 203)(31 61 162 222)(32 80 163 181)(33 99 164 200)(34 118 165 219)(35 77 166 238)(36 96 167 197)(37 115 168 216)(38 74 169 235)(39 93 170 194)(40 112 171 213)(41 71 172 232)(42 90 173 191)(43 109 174 210)(44 68 175 229)(45 87 176 188)(46 106 177 207)(47 65 178 226)(48 84 179 185)(49 103 180 204)(50 62 121 223)(51 81 122 182)(52 100 123 201)(53 119 124 220)(54 78 125 239)(55 97 126 198)(56 116 127 217)(57 75 128 236)(58 94 129 195)(59 113 130 214)(60 72 131 233)
(1 222 31 192)(2 221 32 191)(3 220 33 190)(4 219 34 189)(5 218 35 188)(6 217 36 187)(7 216 37 186)(8 215 38 185)(9 214 39 184)(10 213 40 183)(11 212 41 182)(12 211 42 181)(13 210 43 240)(14 209 44 239)(15 208 45 238)(16 207 46 237)(17 206 47 236)(18 205 48 235)(19 204 49 234)(20 203 50 233)(21 202 51 232)(22 201 52 231)(23 200 53 230)(24 199 54 229)(25 198 55 228)(26 197 56 227)(27 196 57 226)(28 195 58 225)(29 194 59 224)(30 193 60 223)(61 162 91 132)(62 161 92 131)(63 160 93 130)(64 159 94 129)(65 158 95 128)(66 157 96 127)(67 156 97 126)(68 155 98 125)(69 154 99 124)(70 153 100 123)(71 152 101 122)(72 151 102 121)(73 150 103 180)(74 149 104 179)(75 148 105 178)(76 147 106 177)(77 146 107 176)(78 145 108 175)(79 144 109 174)(80 143 110 173)(81 142 111 172)(82 141 112 171)(83 140 113 170)(84 139 114 169)(85 138 115 168)(86 137 116 167)(87 136 117 166)(88 135 118 165)(89 134 119 164)(90 133 120 163)

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

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

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

72 conjugacy classes

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

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + - + + + + + + + - - - + + + - - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D5 D6 D6 C4○D4 D10 D10 D10 D12 C5⋊D4 Dic10 C4○D20 D4⋊2S3 S3×Q8 S3×D5 C5⋊D12 C2×S3×D5 D20⋊5S3 S3×Dic10 kernel C60.45D4 D6⋊Dic5 C30.Q8 C3×C4⋊Dic5 S3×C2×C20 C2×Dic30 C4⋊Dic5 C60 S3×C10 S3×C2×C4 C2×Dic5 C2×C20 C30 C2×Dic3 C2×C12 C22×S3 C20 C12 D6 C6 C10 C10 C2×C4 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 2 2 2 2 1 2 2 2 2 4 8 8 8 1 1 2 4 2 4 4

Matrix representation of C60.45D4 in GL6(𝔽61)

 0 60 0 0 0 0 1 1 0 0 0 0 0 0 8 0 0 0 0 0 11 23 0 0 0 0 0 0 52 0 0 0 0 0 27 27
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 3 54 0 0 0 0 36 58 0 0 0 0 0 0 48 3 0 0 0 0 45 13
,
 60 0 0 0 0 0 1 1 0 0 0 0 0 0 3 54 0 0 0 0 45 58 0 0 0 0 0 0 48 3 0 0 0 0 5 13

G:=sub<GL(6,GF(61))| [0,1,0,0,0,0,60,1,0,0,0,0,0,0,8,11,0,0,0,0,0,23,0,0,0,0,0,0,52,27,0,0,0,0,0,27],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,3,36,0,0,0,0,54,58,0,0,0,0,0,0,48,45,0,0,0,0,3,13],[60,1,0,0,0,0,0,1,0,0,0,0,0,0,3,45,0,0,0,0,54,58,0,0,0,0,0,0,48,5,0,0,0,0,3,13] >;

C60.45D4 in GAP, Magma, Sage, TeX

C_{60}._{45}D_4
% in TeX

G:=Group("C60.45D4");
// GroupNames label

G:=SmallGroup(480,441);
// by ID

G=gap.SmallGroup(480,441);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,253,590,142,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;
// generators/relations

׿
×
𝔽