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G = C30.D5order 300 = 22·3·52

3rd non-split extension by C30 of D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, A-group

Aliases: C30.3D5, C153Dic5, C52Dic15, C10.3D15, C528Dic3, (C5×C15)⋊9C4, C6.(C5⋊D5), C2.(C5⋊D15), C3⋊(C526C4), (C5×C10).3S3, (C5×C30).1C2, SmallGroup(300,20)

Series: Derived Chief Lower central Upper central

C1C5×C15 — C30.D5
C1C5C52C5×C15C5×C30 — C30.D5
C5×C15 — C30.D5
C1C2

Generators and relations for C30.D5
 G = < a,b,c | a30=b5=1, c2=a15, ab=ba, cac-1=a-1, cbc-1=b-1 >

75C4
25Dic3
15Dic5
15Dic5
15Dic5
15Dic5
15Dic5
15Dic5
5Dic15
5Dic15
5Dic15
5Dic15
5Dic15
5Dic15
3C526C4

Smallest permutation representation of C30.D5
Regular action on 300 points
Generators in S300
(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)(241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270)(271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300)
(1 254 59 297 225)(2 255 60 298 226)(3 256 31 299 227)(4 257 32 300 228)(5 258 33 271 229)(6 259 34 272 230)(7 260 35 273 231)(8 261 36 274 232)(9 262 37 275 233)(10 263 38 276 234)(11 264 39 277 235)(12 265 40 278 236)(13 266 41 279 237)(14 267 42 280 238)(15 268 43 281 239)(16 269 44 282 240)(17 270 45 283 211)(18 241 46 284 212)(19 242 47 285 213)(20 243 48 286 214)(21 244 49 287 215)(22 245 50 288 216)(23 246 51 289 217)(24 247 52 290 218)(25 248 53 291 219)(26 249 54 292 220)(27 250 55 293 221)(28 251 56 294 222)(29 252 57 295 223)(30 253 58 296 224)(61 156 110 139 195)(62 157 111 140 196)(63 158 112 141 197)(64 159 113 142 198)(65 160 114 143 199)(66 161 115 144 200)(67 162 116 145 201)(68 163 117 146 202)(69 164 118 147 203)(70 165 119 148 204)(71 166 120 149 205)(72 167 91 150 206)(73 168 92 121 207)(74 169 93 122 208)(75 170 94 123 209)(76 171 95 124 210)(77 172 96 125 181)(78 173 97 126 182)(79 174 98 127 183)(80 175 99 128 184)(81 176 100 129 185)(82 177 101 130 186)(83 178 102 131 187)(84 179 103 132 188)(85 180 104 133 189)(86 151 105 134 190)(87 152 106 135 191)(88 153 107 136 192)(89 154 108 137 193)(90 155 109 138 194)
(1 108 16 93)(2 107 17 92)(3 106 18 91)(4 105 19 120)(5 104 20 119)(6 103 21 118)(7 102 22 117)(8 101 23 116)(9 100 24 115)(10 99 25 114)(11 98 26 113)(12 97 27 112)(13 96 28 111)(14 95 29 110)(15 94 30 109)(31 87 46 72)(32 86 47 71)(33 85 48 70)(34 84 49 69)(35 83 50 68)(36 82 51 67)(37 81 52 66)(38 80 53 65)(39 79 54 64)(40 78 55 63)(41 77 56 62)(42 76 57 61)(43 75 58 90)(44 74 59 89)(45 73 60 88)(121 226 136 211)(122 225 137 240)(123 224 138 239)(124 223 139 238)(125 222 140 237)(126 221 141 236)(127 220 142 235)(128 219 143 234)(129 218 144 233)(130 217 145 232)(131 216 146 231)(132 215 147 230)(133 214 148 229)(134 213 149 228)(135 212 150 227)(151 242 166 257)(152 241 167 256)(153 270 168 255)(154 269 169 254)(155 268 170 253)(156 267 171 252)(157 266 172 251)(158 265 173 250)(159 264 174 249)(160 263 175 248)(161 262 176 247)(162 261 177 246)(163 260 178 245)(164 259 179 244)(165 258 180 243)(181 294 196 279)(182 293 197 278)(183 292 198 277)(184 291 199 276)(185 290 200 275)(186 289 201 274)(187 288 202 273)(188 287 203 272)(189 286 204 271)(190 285 205 300)(191 284 206 299)(192 283 207 298)(193 282 208 297)(194 281 209 296)(195 280 210 295)

G:=sub<Sym(300)| (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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300), (1,254,59,297,225)(2,255,60,298,226)(3,256,31,299,227)(4,257,32,300,228)(5,258,33,271,229)(6,259,34,272,230)(7,260,35,273,231)(8,261,36,274,232)(9,262,37,275,233)(10,263,38,276,234)(11,264,39,277,235)(12,265,40,278,236)(13,266,41,279,237)(14,267,42,280,238)(15,268,43,281,239)(16,269,44,282,240)(17,270,45,283,211)(18,241,46,284,212)(19,242,47,285,213)(20,243,48,286,214)(21,244,49,287,215)(22,245,50,288,216)(23,246,51,289,217)(24,247,52,290,218)(25,248,53,291,219)(26,249,54,292,220)(27,250,55,293,221)(28,251,56,294,222)(29,252,57,295,223)(30,253,58,296,224)(61,156,110,139,195)(62,157,111,140,196)(63,158,112,141,197)(64,159,113,142,198)(65,160,114,143,199)(66,161,115,144,200)(67,162,116,145,201)(68,163,117,146,202)(69,164,118,147,203)(70,165,119,148,204)(71,166,120,149,205)(72,167,91,150,206)(73,168,92,121,207)(74,169,93,122,208)(75,170,94,123,209)(76,171,95,124,210)(77,172,96,125,181)(78,173,97,126,182)(79,174,98,127,183)(80,175,99,128,184)(81,176,100,129,185)(82,177,101,130,186)(83,178,102,131,187)(84,179,103,132,188)(85,180,104,133,189)(86,151,105,134,190)(87,152,106,135,191)(88,153,107,136,192)(89,154,108,137,193)(90,155,109,138,194), (1,108,16,93)(2,107,17,92)(3,106,18,91)(4,105,19,120)(5,104,20,119)(6,103,21,118)(7,102,22,117)(8,101,23,116)(9,100,24,115)(10,99,25,114)(11,98,26,113)(12,97,27,112)(13,96,28,111)(14,95,29,110)(15,94,30,109)(31,87,46,72)(32,86,47,71)(33,85,48,70)(34,84,49,69)(35,83,50,68)(36,82,51,67)(37,81,52,66)(38,80,53,65)(39,79,54,64)(40,78,55,63)(41,77,56,62)(42,76,57,61)(43,75,58,90)(44,74,59,89)(45,73,60,88)(121,226,136,211)(122,225,137,240)(123,224,138,239)(124,223,139,238)(125,222,140,237)(126,221,141,236)(127,220,142,235)(128,219,143,234)(129,218,144,233)(130,217,145,232)(131,216,146,231)(132,215,147,230)(133,214,148,229)(134,213,149,228)(135,212,150,227)(151,242,166,257)(152,241,167,256)(153,270,168,255)(154,269,169,254)(155,268,170,253)(156,267,171,252)(157,266,172,251)(158,265,173,250)(159,264,174,249)(160,263,175,248)(161,262,176,247)(162,261,177,246)(163,260,178,245)(164,259,179,244)(165,258,180,243)(181,294,196,279)(182,293,197,278)(183,292,198,277)(184,291,199,276)(185,290,200,275)(186,289,201,274)(187,288,202,273)(188,287,203,272)(189,286,204,271)(190,285,205,300)(191,284,206,299)(192,283,207,298)(193,282,208,297)(194,281,209,296)(195,280,210,295)>;

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)(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270)(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300), (1,254,59,297,225)(2,255,60,298,226)(3,256,31,299,227)(4,257,32,300,228)(5,258,33,271,229)(6,259,34,272,230)(7,260,35,273,231)(8,261,36,274,232)(9,262,37,275,233)(10,263,38,276,234)(11,264,39,277,235)(12,265,40,278,236)(13,266,41,279,237)(14,267,42,280,238)(15,268,43,281,239)(16,269,44,282,240)(17,270,45,283,211)(18,241,46,284,212)(19,242,47,285,213)(20,243,48,286,214)(21,244,49,287,215)(22,245,50,288,216)(23,246,51,289,217)(24,247,52,290,218)(25,248,53,291,219)(26,249,54,292,220)(27,250,55,293,221)(28,251,56,294,222)(29,252,57,295,223)(30,253,58,296,224)(61,156,110,139,195)(62,157,111,140,196)(63,158,112,141,197)(64,159,113,142,198)(65,160,114,143,199)(66,161,115,144,200)(67,162,116,145,201)(68,163,117,146,202)(69,164,118,147,203)(70,165,119,148,204)(71,166,120,149,205)(72,167,91,150,206)(73,168,92,121,207)(74,169,93,122,208)(75,170,94,123,209)(76,171,95,124,210)(77,172,96,125,181)(78,173,97,126,182)(79,174,98,127,183)(80,175,99,128,184)(81,176,100,129,185)(82,177,101,130,186)(83,178,102,131,187)(84,179,103,132,188)(85,180,104,133,189)(86,151,105,134,190)(87,152,106,135,191)(88,153,107,136,192)(89,154,108,137,193)(90,155,109,138,194), (1,108,16,93)(2,107,17,92)(3,106,18,91)(4,105,19,120)(5,104,20,119)(6,103,21,118)(7,102,22,117)(8,101,23,116)(9,100,24,115)(10,99,25,114)(11,98,26,113)(12,97,27,112)(13,96,28,111)(14,95,29,110)(15,94,30,109)(31,87,46,72)(32,86,47,71)(33,85,48,70)(34,84,49,69)(35,83,50,68)(36,82,51,67)(37,81,52,66)(38,80,53,65)(39,79,54,64)(40,78,55,63)(41,77,56,62)(42,76,57,61)(43,75,58,90)(44,74,59,89)(45,73,60,88)(121,226,136,211)(122,225,137,240)(123,224,138,239)(124,223,139,238)(125,222,140,237)(126,221,141,236)(127,220,142,235)(128,219,143,234)(129,218,144,233)(130,217,145,232)(131,216,146,231)(132,215,147,230)(133,214,148,229)(134,213,149,228)(135,212,150,227)(151,242,166,257)(152,241,167,256)(153,270,168,255)(154,269,169,254)(155,268,170,253)(156,267,171,252)(157,266,172,251)(158,265,173,250)(159,264,174,249)(160,263,175,248)(161,262,176,247)(162,261,177,246)(163,260,178,245)(164,259,179,244)(165,258,180,243)(181,294,196,279)(182,293,197,278)(183,292,198,277)(184,291,199,276)(185,290,200,275)(186,289,201,274)(187,288,202,273)(188,287,203,272)(189,286,204,271)(190,285,205,300)(191,284,206,299)(192,283,207,298)(193,282,208,297)(194,281,209,296)(195,280,210,295) );

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),(241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270),(271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300)], [(1,254,59,297,225),(2,255,60,298,226),(3,256,31,299,227),(4,257,32,300,228),(5,258,33,271,229),(6,259,34,272,230),(7,260,35,273,231),(8,261,36,274,232),(9,262,37,275,233),(10,263,38,276,234),(11,264,39,277,235),(12,265,40,278,236),(13,266,41,279,237),(14,267,42,280,238),(15,268,43,281,239),(16,269,44,282,240),(17,270,45,283,211),(18,241,46,284,212),(19,242,47,285,213),(20,243,48,286,214),(21,244,49,287,215),(22,245,50,288,216),(23,246,51,289,217),(24,247,52,290,218),(25,248,53,291,219),(26,249,54,292,220),(27,250,55,293,221),(28,251,56,294,222),(29,252,57,295,223),(30,253,58,296,224),(61,156,110,139,195),(62,157,111,140,196),(63,158,112,141,197),(64,159,113,142,198),(65,160,114,143,199),(66,161,115,144,200),(67,162,116,145,201),(68,163,117,146,202),(69,164,118,147,203),(70,165,119,148,204),(71,166,120,149,205),(72,167,91,150,206),(73,168,92,121,207),(74,169,93,122,208),(75,170,94,123,209),(76,171,95,124,210),(77,172,96,125,181),(78,173,97,126,182),(79,174,98,127,183),(80,175,99,128,184),(81,176,100,129,185),(82,177,101,130,186),(83,178,102,131,187),(84,179,103,132,188),(85,180,104,133,189),(86,151,105,134,190),(87,152,106,135,191),(88,153,107,136,192),(89,154,108,137,193),(90,155,109,138,194)], [(1,108,16,93),(2,107,17,92),(3,106,18,91),(4,105,19,120),(5,104,20,119),(6,103,21,118),(7,102,22,117),(8,101,23,116),(9,100,24,115),(10,99,25,114),(11,98,26,113),(12,97,27,112),(13,96,28,111),(14,95,29,110),(15,94,30,109),(31,87,46,72),(32,86,47,71),(33,85,48,70),(34,84,49,69),(35,83,50,68),(36,82,51,67),(37,81,52,66),(38,80,53,65),(39,79,54,64),(40,78,55,63),(41,77,56,62),(42,76,57,61),(43,75,58,90),(44,74,59,89),(45,73,60,88),(121,226,136,211),(122,225,137,240),(123,224,138,239),(124,223,139,238),(125,222,140,237),(126,221,141,236),(127,220,142,235),(128,219,143,234),(129,218,144,233),(130,217,145,232),(131,216,146,231),(132,215,147,230),(133,214,148,229),(134,213,149,228),(135,212,150,227),(151,242,166,257),(152,241,167,256),(153,270,168,255),(154,269,169,254),(155,268,170,253),(156,267,171,252),(157,266,172,251),(158,265,173,250),(159,264,174,249),(160,263,175,248),(161,262,176,247),(162,261,177,246),(163,260,178,245),(164,259,179,244),(165,258,180,243),(181,294,196,279),(182,293,197,278),(183,292,198,277),(184,291,199,276),(185,290,200,275),(186,289,201,274),(187,288,202,273),(188,287,203,272),(189,286,204,271),(190,285,205,300),(191,284,206,299),(192,283,207,298),(193,282,208,297),(194,281,209,296),(195,280,210,295)])

78 conjugacy classes

class 1  2  3 4A4B5A···5L 6 10A···10L15A···15X30A···30X
order123445···5610···1015···1530···30
size11275752···222···22···22···2

78 irreducible representations

dim111222222
type++++--+-
imageC1C2C4S3D5Dic3Dic5D15Dic15
kernelC30.D5C5×C30C5×C15C5×C10C30C52C15C10C5
# reps1121121122424

Matrix representation of C30.D5 in GL4(𝔽61) generated by

44100
166000
001430
003136
,
44100
166000
0001
006017
,
205900
474100
002453
003437
G:=sub<GL(4,GF(61))| [44,16,0,0,1,60,0,0,0,0,14,31,0,0,30,36],[44,16,0,0,1,60,0,0,0,0,0,60,0,0,1,17],[20,47,0,0,59,41,0,0,0,0,24,34,0,0,53,37] >;

C30.D5 in GAP, Magma, Sage, TeX

C_{30}.D_5
% in TeX

G:=Group("C30.D5");
// GroupNames label

G:=SmallGroup(300,20);
// by ID

G=gap.SmallGroup(300,20);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,10,122,963,6004]);
// Polycyclic

G:=Group<a,b,c|a^30=b^5=1,c^2=a^15,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C30.D5 in TeX

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