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G = C60.17D4order 480 = 25·3·5

17th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.17D4, C23.12D30, (C6×D4).5D5, (D4×C30).5C2, (C2×D4).6D15, (D4×C10).5S3, (C2×C4).50D30, (C4×Dic15)⋊5C2, (C2×C20).148D6, C30.382(C2×D4), C4.7(C157D4), (C2×Dic30)⋊13C2, C30.38D49C2, (C2×C12).147D10, C54(C23.12D6), C20.42(C3⋊D4), C34(C20.17D4), C1522(C4.4D4), C12.44(C5⋊D4), (C2×C60).74C22, (C22×C6).64D10, (C22×C10).79D6, C30.224(C4○D4), (C2×C30).307C23, C2.16(D42D15), C6.103(D42D5), (C22×C30).20C22, C10.103(D42S3), C22.58(C22×D15), (C2×Dic15).17C22, C6.107(C2×C5⋊D4), C2.12(C2×C157D4), C10.107(C2×C3⋊D4), (C2×C6).303(C22×D5), (C2×C10).302(C22×S3), SmallGroup(480,901)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.17D4
C1C5C15C30C2×C30C2×Dic15C4×Dic15 — C60.17D4
C15C2×C30 — C60.17D4
C1C22C2×D4

Generators and relations for C60.17D4
 G = < a,b,c | a60=b4=1, c2=a30, bab-1=a29, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 692 in 152 conjugacy classes, 55 normal (23 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], C10, C10 [×2], C10 [×2], Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×6], C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], Dic6 [×2], C2×Dic3 [×4], C2×C12, C3×D4 [×2], C22×C6 [×2], C30, C30 [×2], C30 [×2], C4.4D4, Dic10 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C4×Dic3, C6.D4 [×4], C2×Dic6, C6×D4, Dic15 [×4], C60 [×2], C2×C30, C2×C30 [×6], C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C23.12D6, Dic30 [×2], C2×Dic15 [×4], C2×C60, D4×C15 [×2], C22×C30 [×2], C20.17D4, C4×Dic15, C30.38D4 [×4], C2×Dic30, D4×C30, C60.17D4
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, D15, C4.4D4, C5⋊D4 [×2], C22×D5, D42S3 [×2], C2×C3⋊D4, D30 [×3], D42D5 [×2], C2×C5⋊D4, C23.12D6, C157D4 [×2], C22×D15, C20.17D4, D42D15 [×2], C2×C157D4, C60.17D4

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

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

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

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

84 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B15C15D20A20B20C20D30A···30L30M···30AB60A···60H
order12222234444444455666666610···1010···101212151515152020202030···3030···3060···60
size1111442223030303060602222244442···24···444222244442···24···44···4

84 irreducible representations

dim1111122222222222222444
type+++++++++++++++---
imageC1C2C2C2C2S3D4D5D6D6C4○D4D10D10C3⋊D4D15C5⋊D4D30D30C157D4D42S3D42D5D42D15
kernelC60.17D4C4×Dic15C30.38D4C2×Dic30D4×C30D4×C10C60C6×D4C2×C20C22×C10C30C2×C12C22×C6C20C2×D4C12C2×C4C23C4C10C6C2
# reps11411122124244484816248

Matrix representation of C60.17D4 in GL6(𝔽61)

6050000
39480000
009000
0063400
00006013
0000281
,
46360000
48150000
00434700
00451800
0000110
0000011
,
46360000
48150000
00181400
00514300
00005021
0000011

G:=sub<GL(6,GF(61))| [60,39,0,0,0,0,5,48,0,0,0,0,0,0,9,6,0,0,0,0,0,34,0,0,0,0,0,0,60,28,0,0,0,0,13,1],[46,48,0,0,0,0,36,15,0,0,0,0,0,0,43,45,0,0,0,0,47,18,0,0,0,0,0,0,11,0,0,0,0,0,0,11],[46,48,0,0,0,0,36,15,0,0,0,0,0,0,18,51,0,0,0,0,14,43,0,0,0,0,0,0,50,0,0,0,0,0,21,11] >;

C60.17D4 in GAP, Magma, Sage, TeX

C_{60}._{17}D_4
% in TeX

G:=Group("C60.17D4");
// GroupNames label

G:=SmallGroup(480,901);
// by ID

G=gap.SmallGroup(480,901);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,64,590,135,2693,18822]);
// Polycyclic

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

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