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G = C60.69D4order 480 = 25·3·5

69th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.69D4, C20.19D12, D6⋊Dic56C2, (C4×Dic5)⋊3S3, (C2×D12).3D5, C54(C427S3), (C12×Dic5)⋊3C2, (C10×D12).4C2, C10.57(C2×D12), C30.115(C2×D4), (C2×C20).115D6, C4.9(C5⋊D12), C157(C4.4D4), (C2×Dic30)⋊28C2, C30.39(C4○D4), (C2×C12).298D10, C12.59(C5⋊D4), C31(C20.17D4), (C2×C30).63C23, (C22×S3).9D10, C10.54(C4○D12), C6.22(D42D5), (C2×C60).142C22, (C2×Dic5).164D6, C2.11(D125D5), (C6×Dic5).186C22, (C2×Dic15).60C22, C6.11(C2×C5⋊D4), (C2×C4).155(S3×D5), C2.15(C2×C5⋊D12), (S3×C2×C10).9C22, C22.149(C2×S3×D5), (C2×C6).75(C22×D5), (C2×C10).75(C22×S3), SmallGroup(480,449)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.69D4
C1C5C15C30C2×C30C6×Dic5D6⋊Dic5 — C60.69D4
C15C2×C30 — C60.69D4
C1C22C2×C4

Generators and relations for C60.69D4
 G = < a,b,c | a60=b4=1, c2=a30, bab-1=a49, cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 716 in 152 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C5, S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C15, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], Dic6 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C22×S3 [×2], C5×S3 [×2], C30, C30 [×2], C4.4D4, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C5×D4 [×2], C22×C10 [×2], D6⋊C4 [×4], C4×C12, C2×Dic6, C2×D12, C3×Dic5 [×2], Dic15 [×2], C60 [×2], S3×C10 [×6], C2×C30, C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C427S3, C6×Dic5 [×2], C5×D12 [×2], Dic30 [×2], C2×Dic15 [×2], C2×C60, S3×C2×C10 [×2], C20.17D4, D6⋊Dic5 [×4], C12×Dic5, C10×D12, C2×Dic30, C60.69D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, C4○D4 [×2], D10 [×3], D12 [×2], C22×S3, C4.4D4, C5⋊D4 [×2], C22×D5, C2×D12, C4○D12 [×2], S3×D5, D42D5 [×2], C2×C5⋊D4, C427S3, C5⋊D12 [×2], C2×S3×D5, C20.17D4, D125D5 [×2], C2×C5⋊D12, C60.69D4

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C10A···10F10G···10N12A12B12C12D12E···12L15A15B20A20B20C20D30A···30F60A···60H
order1222223444444445566610···1010···101212121212···1215152020202030···3060···60
size11111212222101010106060222222···212···12222210···104444444···44···4

66 irreducible representations

dim111112222222222244444
type++++++++++++++-++-
imageC1C2C2C2C2S3D4D5D6D6C4○D4D10D10D12C5⋊D4C4○D12S3×D5D42D5C5⋊D12C2×S3×D5D125D5
kernelC60.69D4D6⋊Dic5C12×Dic5C10×D12C2×Dic30C4×Dic5C60C2×D12C2×Dic5C2×C20C30C2×C12C22×S3C20C12C10C2×C4C6C4C22C2
# reps141111222142448824428

Matrix representation of C60.69D4 in GL6(𝔽61)

60460000
5310000
009000
0023400
00006019
0000482
,
50180000
34110000
00345900
00602700
000010
000001
,
5000000
34110000
0027200
0023400
0000600
0000481

G:=sub<GL(6,GF(61))| [60,53,0,0,0,0,46,1,0,0,0,0,0,0,9,2,0,0,0,0,0,34,0,0,0,0,0,0,60,48,0,0,0,0,19,2],[50,34,0,0,0,0,18,11,0,0,0,0,0,0,34,60,0,0,0,0,59,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,34,0,0,0,0,0,11,0,0,0,0,0,0,27,2,0,0,0,0,2,34,0,0,0,0,0,0,60,48,0,0,0,0,0,1] >;

C60.69D4 in GAP, Magma, Sage, TeX

C_{60}._{69}D_4
% in TeX

G:=Group("C60.69D4");
// GroupNames label

G:=SmallGroup(480,449);
// by ID

G=gap.SmallGroup(480,449);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,365,219,100,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=a^49,c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;
// generators/relations

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