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G = C40.69D6order 480 = 25·3·5

5th non-split extension by C40 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.69D6, C4.20D60, C8.16D30, D12015C2, C60.168D4, C20.38D12, C12.38D20, C24.69D10, C22.1D60, Dic6015C2, C120.81C22, C60.246C23, D60.36C22, Dic30.37C22, (C2×C40)⋊6S3, (C2×C24)⋊6D5, (C2×C8)⋊4D15, C54(C4○D24), (C2×C120)⋊10C2, C1525(C4○D8), (C2×C4).82D30, C2.13(C2×D60), (C2×C6).20D20, C6.39(C2×D20), C24⋊D515C2, C34(D407C2), (C2×C20).400D6, (C2×C10).20D12, C10.40(C2×D12), (C2×C30).105D4, C30.267(C2×D4), D6011C21C2, (C2×C12).405D10, C4.27(C22×D15), C20.217(C22×S3), (C2×C60).486C22, C12.219(C22×D5), SmallGroup(480,869)

Series: Derived Chief Lower central Upper central

C1C60 — C40.69D6
C1C5C15C30C60D60D6011C2 — C40.69D6
C15C30C60 — C40.69D6
C1C4C2×C4C2×C8

Generators and relations for C40.69D6
 G = < a,b,c | a4=1, b60=c2=a2, ab=ba, ac=ca, cbc-1=b59 >

Subgroups: 884 in 124 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D5 [×2], C10, C10, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic5 [×2], C20 [×2], D10 [×2], C2×C10, C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, D15 [×2], C30, C30, C4○D8, C40 [×2], Dic10 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C2×C20, C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12 [×2], Dic15 [×2], C60 [×2], D30 [×2], C2×C30, C40⋊C2 [×2], D40, Dic20, C2×C40, C4○D20 [×2], C4○D24, C120 [×2], Dic30 [×2], C4×D15 [×2], D60 [×2], C157D4 [×2], C2×C60, D407C2, C24⋊D5 [×2], D120, Dic60, C2×C120, D6011C2 [×2], C40.69D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C4○D8, D20 [×2], C22×D5, C2×D12, D30 [×3], C2×D20, C4○D24, D60 [×2], C22×D15, D407C2, C2×D60, C40.69D6

Smallest permutation representation of C40.69D6
On 240 points
Generators in S240
(1 31 61 91)(2 32 62 92)(3 33 63 93)(4 34 64 94)(5 35 65 95)(6 36 66 96)(7 37 67 97)(8 38 68 98)(9 39 69 99)(10 40 70 100)(11 41 71 101)(12 42 72 102)(13 43 73 103)(14 44 74 104)(15 45 75 105)(16 46 76 106)(17 47 77 107)(18 48 78 108)(19 49 79 109)(20 50 80 110)(21 51 81 111)(22 52 82 112)(23 53 83 113)(24 54 84 114)(25 55 85 115)(26 56 86 116)(27 57 87 117)(28 58 88 118)(29 59 89 119)(30 60 90 120)(121 211 181 151)(122 212 182 152)(123 213 183 153)(124 214 184 154)(125 215 185 155)(126 216 186 156)(127 217 187 157)(128 218 188 158)(129 219 189 159)(130 220 190 160)(131 221 191 161)(132 222 192 162)(133 223 193 163)(134 224 194 164)(135 225 195 165)(136 226 196 166)(137 227 197 167)(138 228 198 168)(139 229 199 169)(140 230 200 170)(141 231 201 171)(142 232 202 172)(143 233 203 173)(144 234 204 174)(145 235 205 175)(146 236 206 176)(147 237 207 177)(148 238 208 178)(149 239 209 179)(150 240 210 180)
(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)
(1 206 61 146)(2 145 62 205)(3 204 63 144)(4 143 64 203)(5 202 65 142)(6 141 66 201)(7 200 67 140)(8 139 68 199)(9 198 69 138)(10 137 70 197)(11 196 71 136)(12 135 72 195)(13 194 73 134)(14 133 74 193)(15 192 75 132)(16 131 76 191)(17 190 77 130)(18 129 78 189)(19 188 79 128)(20 127 80 187)(21 186 81 126)(22 125 82 185)(23 184 83 124)(24 123 84 183)(25 182 85 122)(26 121 86 181)(27 180 87 240)(28 239 88 179)(29 178 89 238)(30 237 90 177)(31 176 91 236)(32 235 92 175)(33 174 93 234)(34 233 94 173)(35 172 95 232)(36 231 96 171)(37 170 97 230)(38 229 98 169)(39 168 99 228)(40 227 100 167)(41 166 101 226)(42 225 102 165)(43 164 103 224)(44 223 104 163)(45 162 105 222)(46 221 106 161)(47 160 107 220)(48 219 108 159)(49 158 109 218)(50 217 110 157)(51 156 111 216)(52 215 112 155)(53 154 113 214)(54 213 114 153)(55 152 115 212)(56 211 116 151)(57 150 117 210)(58 209 118 149)(59 148 119 208)(60 207 120 147)

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

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

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

126 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B6C8A8B8C8D10A···10F12A12B12C12D15A15B15C15D20A···20H24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222234444455666888810···10121212121515151520···2024···2430···3040···4060···60120···120
size1126060211260602222222222···2222222222···22···22···22···22···22···2

126 irreducible representations

dim111111222222222222222222222
type+++++++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12D12D15C4○D8D20D20D30D30C4○D24D60D60D407C2C40.69D6
kernelC40.69D6C24⋊D5D120Dic60C2×C120D6011C2C2×C40C60C2×C30C2×C24C40C2×C20C24C2×C12C20C2×C10C2×C8C15C12C2×C6C8C2×C4C5C4C22C3C1
# reps12111211122142224444848881632

Matrix representation of C40.69D6 in GL2(𝔽241) generated by

640
064
,
1330
0212
,
0212
1330
G:=sub<GL(2,GF(241))| [64,0,0,64],[133,0,0,212],[0,133,212,0] >;

C40.69D6 in GAP, Magma, Sage, TeX

C_{40}._{69}D_6
% in TeX

G:=Group("C40.69D6");
// GroupNames label

G:=SmallGroup(480,869);
// by ID

G=gap.SmallGroup(480,869);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,58,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^60=c^2=a^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^59>;
// generators/relations

׿
×
𝔽