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G = C60.205D4order 480 = 25·3·5

5th non-split extension by C60 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.205D4, C223Dic30, C23.25D30, (C2×C30)⋊9Q8, C605C414C2, (C2×C6)⋊8Dic10, (C2×C4).84D30, C30.71(C2×Q8), (C2×Dic30)⋊9C2, (C2×C10)⋊11Dic6, C30.375(C2×D4), (C2×C20).393D6, (C22×C60).9C2, C1534(C22⋊Q8), C30.4Q82C2, C2.9(C2×Dic30), (C22×C4).7D15, (C22×C12).6D5, (C2×C12).382D10, C55(C12.48D4), C4.23(C157D4), C35(C20.48D4), (C22×C20).10S3, C10.39(C2×Dic6), C6.39(C2×Dic10), C6.103(C4○D20), C30.175(C4○D4), C12.102(C5⋊D4), C20.102(C3⋊D4), (C2×C30).298C23, (C2×C60).464C22, C30.38D4.4C2, (C22×C10).133D6, (C22×C6).115D10, C10.103(C4○D12), C2.17(D6011C2), C22.54(C22×D15), (C22×C30).138C22, (C2×Dic15).13C22, C6.98(C2×C5⋊D4), C2.5(C2×C157D4), C10.98(C2×C3⋊D4), (C2×C6).294(C22×D5), (C2×C10).293(C22×S3), SmallGroup(480,889)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.205D4
C1C5C15C30C2×C30C2×Dic15C2×Dic30 — C60.205D4
C15C2×C30 — C60.205D4
C1C22C22×C4

Generators and relations for C60.205D4
 G = < a,b,c | a60=b4=1, c2=a30, bab-1=cac-1=a-1, cbc-1=a30b-1 >

Subgroups: 660 in 148 conjugacy classes, 63 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C10 [×3], C10 [×2], Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C30 [×3], C30 [×2], C22⋊Q8, Dic10 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, Dic15 [×4], C60 [×2], C60, C2×C30, C2×C30 [×2], C2×C30 [×2], C10.D4 [×2], C4⋊Dic5, C23.D5 [×2], C2×Dic10, C22×C20, C12.48D4, Dic30 [×2], C2×Dic15 [×4], C2×C60 [×2], C2×C60 [×2], C22×C30, C20.48D4, C30.4Q8 [×2], C605C4, C30.38D4 [×2], C2×Dic30, C22×C60, C60.205D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D5, D6 [×3], C2×D4, C2×Q8, C4○D4, D10 [×3], Dic6 [×2], C3⋊D4 [×2], C22×S3, D15, C22⋊Q8, Dic10 [×2], C5⋊D4 [×2], C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, D30 [×3], C2×Dic10, C4○D20, C2×C5⋊D4, C12.48D4, Dic30 [×2], C157D4 [×2], C22×D15, C20.48D4, C2×Dic30, D6011C2, C2×C157D4, C60.205D4

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

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

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

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

126 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A···6G10A···10N12A···12H15A15B15C15D20A···20P30A···30AB60A···60AF
order122222344444444556···610···1012···121515151520···2030···3060···60
size1111222222260606060222···22···22···222222···22···22···2

126 irreducible representations

dim111111222222222222222222222
type++++++++-+++++-+-++-
imageC1C2C2C2C2C2S3D4Q8D5D6D6C4○D4D10D10C3⋊D4Dic6D15C5⋊D4Dic10C4○D12D30D30C4○D20C157D4Dic30D6011C2
kernelC60.205D4C30.4Q8C605C4C30.38D4C2×Dic30C22×C60C22×C20C60C2×C30C22×C12C2×C20C22×C10C30C2×C12C22×C6C20C2×C10C22×C4C12C2×C6C10C2×C4C23C6C4C22C2
# reps121211122221242444884848161616

Matrix representation of C60.205D4 in GL4(𝔽61) generated by

50000
01100
0050
00049
,
0100
1000
0001
00600
,
0100
60000
0001
0010
G:=sub<GL(4,GF(61))| [50,0,0,0,0,11,0,0,0,0,5,0,0,0,0,49],[0,1,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[0,60,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C60.205D4 in GAP, Magma, Sage, TeX

C_{60}._{205}D_4
% in TeX

G:=Group("C60.205D4");
// GroupNames label

G:=SmallGroup(480,889);
// by ID

G=gap.SmallGroup(480,889);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,254,2693,18822]);
// Polycyclic

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

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