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G = C60.88D4order 480 = 25·3·5

88th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.88D4, (C6×D20).8C2, (C2×D20).9S3, (C2×Dic6)⋊10D5, (C2×C20).113D6, C30.112(C2×D4), C4.9(C15⋊D4), C155(C4.4D4), (C4×Dic15)⋊23C2, (C10×Dic6)⋊10C2, C30.34(C4○D4), (C2×C12).114D10, D10⋊Dic36C2, C12.36(C5⋊D4), C20.35(C3⋊D4), C33(C20.23D4), C53(C23.12D6), (C2×C30).58C23, (C22×D5).11D6, C10.7(D42S3), (C2×C60).194C22, C6.27(Q82D5), (C2×Dic3).16D10, C2.11(D20⋊S3), (C10×Dic3).35C22, (C2×Dic15).188C22, C6.81(C2×C5⋊D4), (D5×C2×C6).9C22, (C2×C4).204(S3×D5), C2.15(C2×C15⋊D4), C10.82(C2×C3⋊D4), C22.145(C2×S3×D5), (C2×C6).70(C22×D5), (C2×C10).70(C22×S3), SmallGroup(480,444)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.88D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — C60.88D4
C15C2×C30 — C60.88D4
C1C22C2×C4

Generators and relations for C60.88D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a19, cbc=a30b-1 >

Subgroups: 732 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C3×D4 [×2], C22×C6 [×2], C3×D5 [×2], C30, C30 [×2], C4.4D4, D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, C5×Dic3 [×2], Dic15 [×2], C60 [×2], C6×D5 [×6], C2×C30, C4×Dic5, D10⋊C4 [×4], C2×D20, Q8×C10, C23.12D6, C3×D20 [×2], C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], C20.23D4, D10⋊Dic3 [×4], C4×Dic15, C6×D20, C10×Dic6, C60.88D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], C3⋊D4 [×2], C22×S3, C4.4D4, C5⋊D4 [×2], C22×D5, D42S3 [×2], C2×C3⋊D4, S3×D5, Q82D5 [×2], C2×C5⋊D4, C23.12D6, C15⋊D4 [×2], C2×S3×D5, C20.23D4, D20⋊S3 [×2], C2×C15⋊D4, C60.88D4

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

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

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

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

60 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F6G10A···10F12A12B15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222234444444455666666610···10121215152020202020···2030···3060···60
size1111202022212123030303022222202020202···24444444412···124···44···4

60 irreducible representations

dim111112222222222444444
type++++++++++++-++-+
imageC1C2C2C2C2S3D4D5D6D6C4○D4D10D10C3⋊D4C5⋊D4D42S3S3×D5Q82D5C15⋊D4C2×S3×D5D20⋊S3
kernelC60.88D4D10⋊Dic3C4×Dic15C6×D20C10×Dic6C2×D20C60C2×Dic6C2×C20C22×D5C30C2×Dic3C2×C12C20C12C10C2×C4C6C4C22C2
# reps141111221244248224428

Matrix representation of C60.88D4 in GL6(𝔽61)

010000
6000000
00444400
00176000
0000130
00004547
,
5000000
0500000
00144500
00394700
0000172
00003844
,
010000
100000
0060000
0044100
0000600
0000171

G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,0,0,0,0,0,0,0,44,17,0,0,0,0,44,60,0,0,0,0,0,0,13,45,0,0,0,0,0,47],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,14,39,0,0,0,0,45,47,0,0,0,0,0,0,17,38,0,0,0,0,2,44],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,60,44,0,0,0,0,0,1,0,0,0,0,0,0,60,17,0,0,0,0,0,1] >;

C60.88D4 in GAP, Magma, Sage, TeX

C_{60}._{88}D_4
% in TeX

G:=Group("C60.88D4");
// GroupNames label

G:=SmallGroup(480,444);
// by ID

G=gap.SmallGroup(480,444);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,120,422,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=a^30*b^-1>;
// generators/relations

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