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G = C8.D30order 480 = 25·3·5

1st non-split extension by C8 of D30 acting via D30/C15=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C40.3D6, C8.1D30, C4.15D60, C60.13D4, C24.3D10, Dic602C2, C20.25D12, C12.25D20, C22.6D60, M4(2)⋊2D15, C120.1C22, C60.249C23, D60.37C22, Dic30.39C22, (C2×C30).6D4, C24⋊D52C2, C6.42(C2×D20), (C2×C6).11D20, C2.16(C2×D60), (C2×C4).15D30, C53(C8.D6), C30.270(C2×D4), (C2×C20).143D6, (C2×C10).11D12, C10.43(C2×D12), C33(C8.D10), (C3×M4(2))⋊2D5, (C5×M4(2))⋊2S3, (C2×Dic30)⋊11C2, (C2×C12).142D10, C1525(C8.C22), (C15×M4(2))⋊2C2, (C2×C60).69C22, C4.30(C22×D15), C20.220(C22×S3), C12.222(C22×D5), D6011C2.10C2, SmallGroup(480,874)

Series: Derived Chief Lower central Upper central

C1C60 — C8.D30
C1C5C15C30C60D60D6011C2 — C8.D30
C15C30C60 — C8.D30
C1C2C2×C4M4(2)

Generators and relations for C8.D30
 G = < a,b,c | a8=1, b30=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b29 >

Subgroups: 788 in 120 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], D5, C10, C10, Dic3 [×3], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5 [×3], C20 [×2], D10, C2×C10, C24 [×2], Dic6 [×4], C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C8.C22, C40 [×2], Dic10 [×4], C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C24⋊C2 [×2], Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, Dic15 [×3], C60 [×2], D30, C2×C30, C40⋊C2 [×2], Dic20 [×2], C5×M4(2), C2×Dic10, C4○D20, C8.D6, C120 [×2], Dic30, Dic30 [×2], Dic30, C4×D15, D60, C2×Dic15, C157D4, C2×C60, C8.D10, C24⋊D5 [×2], Dic60 [×2], C15×M4(2), C2×Dic30, D6011C2, C8.D30
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D5, D6 [×3], C2×D4, D10 [×3], D12 [×2], C22×S3, D15, C8.C22, D20 [×2], C22×D5, C2×D12, D30 [×3], C2×D20, C8.D6, D60 [×2], C22×D15, C8.D10, C2×D60, C8.D30

Smallest permutation representation of C8.D30
On 240 points
Generators in S240
(1 86 16 71 31 116 46 101)(2 117 17 102 32 87 47 72)(3 88 18 73 33 118 48 103)(4 119 19 104 34 89 49 74)(5 90 20 75 35 120 50 105)(6 61 21 106 36 91 51 76)(7 92 22 77 37 62 52 107)(8 63 23 108 38 93 53 78)(9 94 24 79 39 64 54 109)(10 65 25 110 40 95 55 80)(11 96 26 81 41 66 56 111)(12 67 27 112 42 97 57 82)(13 98 28 83 43 68 58 113)(14 69 29 114 44 99 59 84)(15 100 30 85 45 70 60 115)(121 206 166 221 151 236 136 191)(122 237 167 192 152 207 137 222)(123 208 168 223 153 238 138 193)(124 239 169 194 154 209 139 224)(125 210 170 225 155 240 140 195)(126 181 171 196 156 211 141 226)(127 212 172 227 157 182 142 197)(128 183 173 198 158 213 143 228)(129 214 174 229 159 184 144 199)(130 185 175 200 160 215 145 230)(131 216 176 231 161 186 146 201)(132 187 177 202 162 217 147 232)(133 218 178 233 163 188 148 203)(134 189 179 204 164 219 149 234)(135 220 180 235 165 190 150 205)
(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)
(1 124 31 154)(2 153 32 123)(3 122 33 152)(4 151 34 121)(5 180 35 150)(6 149 36 179)(7 178 37 148)(8 147 38 177)(9 176 39 146)(10 145 40 175)(11 174 41 144)(12 143 42 173)(13 172 43 142)(14 141 44 171)(15 170 45 140)(16 139 46 169)(17 168 47 138)(18 137 48 167)(19 166 49 136)(20 135 50 165)(21 164 51 134)(22 133 52 163)(23 162 53 132)(24 131 54 161)(25 160 55 130)(26 129 56 159)(27 158 57 128)(28 127 58 157)(29 156 59 126)(30 125 60 155)(61 219 91 189)(62 188 92 218)(63 217 93 187)(64 186 94 216)(65 215 95 185)(66 184 96 214)(67 213 97 183)(68 182 98 212)(69 211 99 181)(70 240 100 210)(71 209 101 239)(72 238 102 208)(73 207 103 237)(74 236 104 206)(75 205 105 235)(76 234 106 204)(77 203 107 233)(78 232 108 202)(79 201 109 231)(80 230 110 200)(81 199 111 229)(82 228 112 198)(83 197 113 227)(84 226 114 196)(85 195 115 225)(86 224 116 194)(87 193 117 223)(88 222 118 192)(89 191 119 221)(90 220 120 190)

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

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

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

81 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D4E5A5B6A6B8A8B10A10B10C10D12A12B12C15A15B15C15D20A20B20C20D20E20F24A24B24C24D30A30B30C30D30E30F30G30H40A···40H60A···60H60I60J60K60L120A···120P
order1222344444556688101010101212121515151520202020202024242424303030303030303040···4060···6060606060120···120
size11260222606060222444224422422222222444444222244444···42···244444···4

81 irreducible representations

dim111111222222222222222224444
type+++++++++++++++++++++++----
imageC1C2C2C2C2C2S3D4D4D5D6D6D10D10D12D12D15D20D20D30D30D60D60C8.C22C8.D6C8.D10C8.D30
kernelC8.D30C24⋊D5Dic60C15×M4(2)C2×Dic30D6011C2C5×M4(2)C60C2×C30C3×M4(2)C40C2×C20C24C2×C12C20C2×C10M4(2)C12C2×C6C8C2×C4C4C22C15C5C3C1
# reps122111111221422244484881248

Matrix representation of C8.D30 in GL4(𝔽241) generated by

1052162115
010119447
112182138239
119171239138
,
19111300
1075700
9414171128
66194113163
,
1257812203
312191219
212204194165
2128163161
G:=sub<GL(4,GF(241))| [105,0,112,119,2,101,182,171,162,194,138,239,115,47,239,138],[191,107,94,66,113,57,141,194,0,0,71,113,0,0,128,163],[125,31,212,212,78,2,204,8,12,191,194,163,203,219,165,161] >;

C8.D30 in GAP, Magma, Sage, TeX

C_8.D_{30}
% in TeX

G:=Group("C8.D30");
// GroupNames label

G:=SmallGroup(480,874);
// by ID

G=gap.SmallGroup(480,874);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,58,675,80,2693,18822]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^30=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^29>;
// generators/relations

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