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G = C40.51D6order 480 = 25·3·5

12nd non-split extension by C40 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.51D6, C155M5(2), C24.58D10, C120.54C22, C3⋊C164D5, (C6×D5).2C8, (C8×D5).2S3, C6.12(C8×D5), C8.37(S3×D5), C33(C80⋊C2), C30.22(C2×C8), D10.1(C3⋊C8), (D5×C12).3C4, (D5×C24).4C2, C153C1611C2, C12.74(C4×D5), C53(C12.C8), C60.138(C2×C4), Dic5.1(C3⋊C8), (C4×D5).2Dic3, (C3×Dic5).2C8, C4.17(D5×Dic3), C52C8.2Dic3, C20.43(C2×Dic3), (C5×C3⋊C16)⋊6C2, C2.3(D5×C3⋊C8), C10.11(C2×C3⋊C8), (C3×C52C8).3C4, SmallGroup(480,10)

Series: Derived Chief Lower central Upper central

C1C30 — C40.51D6
C1C5C15C30C60C120D5×C24 — C40.51D6
C15C30 — C40.51D6
C1C8

Generators and relations for C40.51D6
 G = < a,b,c | a40=b6=1, c2=a5, bab-1=cac-1=a9, cbc-1=a20b-1 >

10C2
5C22
5C4
10C6
2D5
5C2×C4
5C8
5C2×C6
5C12
2C3×D5
3C16
5C2×C8
15C16
5C24
5C2×C12
15M5(2)
5C3⋊C16
5C2×C24
3C80
3C52C16
5C12.C8
3C80⋊C2

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

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

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

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

84 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B6C8A8B8C8D8E8F10A10B12A12B12C12D15A15B16A16B16C16D16E16F16G16H20A20B20C20D24A24B24C24D24E24F24G24H30A30B40A···40H60A60B60C60D80A···80P120A···120H
order12234445566688888810101212121215151616161616161616202020202424242424242424303040···406060606080···80120···120
size11102111022210101111101022221010446666303030302222222210101010442···244446···64···4

84 irreducible representations

dim1111111122222222222224444
type++++++-+-++-
imageC1C2C2C2C4C4C8C8S3D5Dic3D6Dic3D10C3⋊C8C3⋊C8M5(2)C4×D5C8×D5C12.C8C80⋊C2S3×D5D5×Dic3D5×C3⋊C8C40.51D6
kernelC40.51D6C5×C3⋊C16C153C16D5×C24C3×C52C8D5×C12C3×Dic5C6×D5C8×D5C3⋊C16C52C8C40C4×D5C24Dic5D10C15C12C6C5C3C8C4C2C1
# reps11112244121112224488162248

Matrix representation of C40.51D6 in GL4(𝔽241) generated by

11013100
11017700
00300
00030
,
15100
024000
00150
006216
,
816700
023300
00148119
00293
G:=sub<GL(4,GF(241))| [110,110,0,0,131,177,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,51,240,0,0,0,0,15,62,0,0,0,16],[8,0,0,0,167,233,0,0,0,0,148,2,0,0,119,93] >;

C40.51D6 in GAP, Magma, Sage, TeX

C_{40}._{51}D_6
% in TeX

G:=Group("C40.51D6");
// GroupNames label

G:=SmallGroup(480,10);
// by ID

G=gap.SmallGroup(480,10);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,36,58,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^40=b^6=1,c^2=a^5,b*a*b^-1=c*a*c^-1=a^9,c*b*c^-1=a^20*b^-1>;
// generators/relations

Export

Subgroup lattice of C40.51D6 in TeX

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