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G = C60.67D4order 480 = 25·3·5

67th non-split extension by C60 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C60.67D4, D106Dic6, (C6×D5)⋊6Q8, C6.27(Q8×D5), (C2×Dic6)⋊2D5, C605C430C2, C153(C22⋊Q8), C30.21(C2×Q8), (C10×Dic6)⋊5C2, C34(D103Q8), (C2×C20).111D6, C30.108(C2×D4), Dic155C45C2, C2.11(D5×Dic6), C10.9(C2×Dic6), C30.26(C4○D4), C10.5(C4○D12), (C2×C12).296D10, C20.33(C3⋊D4), C12.84(C5⋊D4), C4.18(C15⋊D4), C54(C12.48D4), (C2×C30).49C23, C6.7(Q82D5), (C22×D5).83D6, (C2×C60).140C22, (C2×Dic5).160D6, (C2×Dic3).10D10, D10⋊Dic3.3C2, C2.10(C12.28D10), (C6×Dic5).182C22, (C2×Dic15).51C22, (C10×Dic3).29C22, (C2×C4×D5).3S3, (D5×C2×C12).3C2, C6.80(C2×C5⋊D4), (C2×C4).153(S3×D5), C2.14(C2×C15⋊D4), C10.81(C2×C3⋊D4), (D5×C2×C6).96C22, C22.136(C2×S3×D5), (C2×C6).61(C22×D5), (C2×C10).61(C22×S3), SmallGroup(480,435)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C60.67D4
C1C5C15C30C2×C30D5×C2×C6D10⋊Dic3 — C60.67D4
C15C2×C30 — C60.67D4
C1C22C2×C4

Generators and relations for C60.67D4
 G = < a,b,c | a60=b4=c2=1, bab-1=a-1, cac=a49, cbc=a30b-1 >

Subgroups: 636 in 148 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×4], C5, C6 [×3], C6 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5 [×2], C10 [×3], Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×4], C15, C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, Dic6 [×2], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×3], C22×C6, C3×D5 [×2], C30 [×3], C22⋊Q8, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], C2×C30, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], C2×C4×D5, Q8×C10, C12.48D4, D5×C12 [×2], C6×Dic5, C5×Dic6 [×2], C10×Dic3 [×2], C2×Dic15 [×2], C2×C60, D5×C2×C6, D103Q8, D10⋊Dic3 [×2], Dic155C4 [×2], C605C4, D5×C2×C12, C10×Dic6, C60.67D4
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, C22⋊Q8, C5⋊D4 [×2], C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, S3×D5, Q8×D5, Q82D5, C2×C5⋊D4, C12.48D4, C15⋊D4 [×2], C2×S3×D5, D103Q8, D5×Dic6, C12.28D10, C2×C15⋊D4, C60.67D4

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H5A5B6A6B6C6D6E6F6G10A···10F12A12B12C12D12E12F12G12H15A15B20A20B20C20D20E···20L30A···30F60A···60H
order12222234444444455666666610···10121212121212121215152020202020···2030···3060···60
size1111101022210101212606022222101010102···222221010101044444412···124···44···4

66 irreducible representations

dim111111222222222222224444444
type++++++++-++++++-+-+-+-+
imageC1C2C2C2C2C2S3D4Q8D5D6D6D6C4○D4D10D10C3⋊D4Dic6C5⋊D4C4○D12S3×D5Q8×D5Q82D5C15⋊D4C2×S3×D5D5×Dic6C12.28D10
kernelC60.67D4D10⋊Dic3Dic155C4C605C4D5×C2×C12C10×Dic6C2×C4×D5C60C6×D5C2×Dic6C2×Dic5C2×C20C22×D5C30C2×Dic3C2×C12C20D10C12C10C2×C4C6C6C4C22C2C2
# reps122111122211124244842224244

Matrix representation of C60.67D4 in GL4(𝔽61) generated by

181800
436000
00290
002540
,
311700
83000
003356
005928
,
1000
436000
0010
00160
G:=sub<GL(4,GF(61))| [18,43,0,0,18,60,0,0,0,0,29,25,0,0,0,40],[31,8,0,0,17,30,0,0,0,0,33,59,0,0,56,28],[1,43,0,0,0,60,0,0,0,0,1,1,0,0,0,60] >;

C60.67D4 in GAP, Magma, Sage, TeX

C_{60}._{67}D_4
% in TeX

G:=Group("C60.67D4");
// GroupNames label

G:=SmallGroup(480,435);
// by ID

G=gap.SmallGroup(480,435);
# by ID

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

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

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