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G = C40.35D6order 480 = 25·3·5

35th non-split extension by C40 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.35D6, C24.38D10, C120.60C22, C60.173C23, C15⋊Q8.5C4, C8⋊S36D5, C8⋊D56S3, C3⋊C8.24D10, D6.2(C4×D5), C8.33(S3×D5), (C8×D15)⋊15C2, C1510(C8○D4), C55(D12.C4), D10.6(C4×S3), (C4×D5).57D6, C52C8.24D6, C5⋊D12.5C4, C3⋊D20.5C4, C15⋊D4.5C4, D30.29(C2×C4), (C4×S3).33D10, Dic3.2(C4×D5), Dic5.6(C4×S3), C32(D20.2C4), C30.39(C22×C4), D30.5C411C2, (S3×C20).33C22, C20.170(C22×S3), C153C8.47C22, Dic15.36(C2×C4), D6.D10.3C2, (D5×C12).57C22, (C4×D15).63C22, C12.170(C22×D5), C6.8(C2×C4×D5), (D5×C3⋊C8)⋊10C2, C2.11(C4×S3×D5), C10.39(S3×C2×C4), C4.143(C2×S3×D5), (C5×C8⋊S3)⋊8C2, (S3×C52C8)⋊10C2, (C6×D5).4(C2×C4), (C3×C8⋊D5)⋊10C2, (C5×C3⋊C8).24C22, (S3×C10).19(C2×C4), (C3×Dic5).4(C2×C4), (C3×C52C8).24C22, (C5×Dic3).20(C2×C4), SmallGroup(480,344)

Series: Derived Chief Lower central Upper central

C1C30 — C40.35D6
C1C5C15C30C60D5×C12D6.D10 — C40.35D6
C15C30 — C40.35D6
C1C4C8

Generators and relations for C40.35D6
 G = < a,b,c | a40=b6=1, c2=a20, bab-1=a29, cac-1=a9, cbc-1=a20b-1 >

Subgroups: 540 in 124 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×2], C6, C6, C8, C8 [×3], C2×C4 [×3], D4 [×3], Q8, D5 [×2], C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4 [×2], C2×C12, C5×S3, C3×D5, D15, C30, C8○D4, C52C8, C52C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4 [×2], C2×C20, S3×C8 [×2], C8⋊S3, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5 [×2], C8⋊D5, C8⋊D5, C2×C52C8, C5×M4(2), C4○D20, D12.C4, C5×C3⋊C8, C3×C52C8, C153C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.2C4, D5×C3⋊C8, S3×C52C8, D30.5C4, C3×C8⋊D5, C5×C8⋊S3, C8×D15, D6.D10, C40.35D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D5, D6 [×3], C22×C4, D10 [×3], C4×S3 [×2], C22×S3, C8○D4, C4×D5 [×2], C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, D12.C4, C2×S3×D5, D20.2C4, C4×S3×D5, C40.35D6

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

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

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

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

66 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B6A6B8A8B8C8D8E8F8G8H8I8J10A10B10C10D12A12B12C15A15B20A20B20C20D20E20F24A24B24C24D30A30B40A40B40C40D40E40F40G40H60A60B60C60D120A···120H
order1222234444455668888888888101010101212121515202020202020242424243030404040404040404060606060120···120
size116103021161030222202266101015151515221212222044222212124420204444441212121244444···4

66 irreducible representations

dim1111111111112222222222222444444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D5D6D6D6D10D10D10C4×S3C4×S3C8○D4C4×D5C4×D5S3×D5D12.C4C2×S3×D5D20.2C4C4×S3×D5C40.35D6
kernelC40.35D6D5×C3⋊C8S3×C52C8D30.5C4C3×C8⋊D5C5×C8⋊S3C8×D15D6.D10C15⋊D4C3⋊D20C5⋊D12C15⋊Q8C8⋊D5C8⋊S3C52C8C40C4×D5C3⋊C8C24C4×S3Dic5D10C15Dic3D6C8C5C4C3C2C1
# reps1111111122221211122222444222448

Matrix representation of C40.35D6 in GL6(𝔽241)

64540000
861770000
0005100
0018918900
00001770
00000177
,
100000
1942400000
0018919000
00535200
00000240
00001240
,
301910000
2062110000
0018919000
00535200
00008981
0000170152

G:=sub<GL(6,GF(241))| [64,86,0,0,0,0,54,177,0,0,0,0,0,0,0,189,0,0,0,0,51,189,0,0,0,0,0,0,177,0,0,0,0,0,0,177],[1,194,0,0,0,0,0,240,0,0,0,0,0,0,189,53,0,0,0,0,190,52,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[30,206,0,0,0,0,191,211,0,0,0,0,0,0,189,53,0,0,0,0,190,52,0,0,0,0,0,0,89,170,0,0,0,0,81,152] >;

C40.35D6 in GAP, Magma, Sage, TeX

C_{40}._{35}D_6
% in TeX

G:=Group("C40.35D6");
// GroupNames label

G:=SmallGroup(480,344);
// by ID

G=gap.SmallGroup(480,344);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,219,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;
// generators/relations

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