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G = C12.59D20order 480 = 25·3·5

13rd non-split extension by C12 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.59D20, C60.107D4, C3⋊C8.1Dic5, (C2×C30).4Q8, C60.98(C2×C4), C30.33(C4⋊C4), C159(C8.C4), C31(C40.6C4), C20.102(C4×S3), (C2×C12).64D10, (C2×C20).310D6, C6.4(C4⋊Dic5), C4.13(S3×Dic5), (C2×C6).2Dic10, (C2×C10).5Dic6, C12.6(C2×Dic5), C4.Dic5.1S3, C22.1(C15⋊Q8), C60.7C4.4C2, C20.64(C3⋊D4), C53(C12.53D4), C4.31(C3⋊D20), (C2×C60).38C22, C2.5(C6.Dic10), C10.15(Dic3⋊C4), (C5×C3⋊C8).5C4, (C2×C3⋊C8).5D5, (C10×C3⋊C8).1C2, (C2×C4).90(S3×D5), (C3×C4.Dic5).2C2, SmallGroup(480,69)

Series: Derived Chief Lower central Upper central

C1C60 — C12.59D20
C1C5C15C30C60C2×C60C3×C4.Dic5 — C12.59D20
C15C30C60 — C12.59D20
C1C4C2×C4

Generators and relations for C12.59D20
 G = < a,b,c | a12=1, b20=a6, c2=a3, bab-1=cac-1=a5, cbc-1=b19 >

Subgroups: 188 in 60 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C8 [×4], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C52C8 [×2], C40 [×2], C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60 [×2], C2×C30, C4.Dic5, C4.Dic5, C2×C40, C12.53D4, C5×C3⋊C8 [×2], C3×C52C8, C153C8, C2×C60, C40.6C4, C3×C4.Dic5, C10×C3⋊C8, C60.7C4, C12.59D20
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, Dic5 [×2], D10, Dic6, C4×S3, C3⋊D4, C8.C4, Dic10, D20, C2×Dic5, Dic3⋊C4, S3×D5, C4⋊Dic5, C12.53D4, S3×Dic5, C3⋊D20, C15⋊Q8, C40.6C4, C6.Dic10, C12.59D20

Smallest permutation representation of C12.59D20
On 240 points
Generators in S240
(1 199 82 11 169 92 21 179 102 31 189 112)(2 93 190 12 103 200 22 113 170 32 83 180)(3 161 84 13 171 94 23 181 104 33 191 114)(4 95 192 14 105 162 24 115 172 34 85 182)(5 163 86 15 173 96 25 183 106 35 193 116)(6 97 194 16 107 164 26 117 174 36 87 184)(7 165 88 17 175 98 27 185 108 37 195 118)(8 99 196 18 109 166 28 119 176 38 89 186)(9 167 90 19 177 100 29 187 110 39 197 120)(10 101 198 20 111 168 30 81 178 40 91 188)(41 236 143 71 226 133 61 216 123 51 206 153)(42 134 207 72 124 237 62 154 227 52 144 217)(43 238 145 73 228 135 63 218 125 53 208 155)(44 136 209 74 126 239 64 156 229 54 146 219)(45 240 147 75 230 137 65 220 127 55 210 157)(46 138 211 76 128 201 66 158 231 56 148 221)(47 202 149 77 232 139 67 222 129 57 212 159)(48 140 213 78 130 203 68 160 233 58 150 223)(49 204 151 79 234 141 69 224 131 59 214 121)(50 142 215 80 132 205 70 122 235 60 152 225)
(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)
(1 240 11 230 21 220 31 210)(2 219 12 209 22 239 32 229)(3 238 13 228 23 218 33 208)(4 217 14 207 24 237 34 227)(5 236 15 226 25 216 35 206)(6 215 16 205 26 235 36 225)(7 234 17 224 27 214 37 204)(8 213 18 203 28 233 38 223)(9 232 19 222 29 212 39 202)(10 211 20 201 30 231 40 221)(41 193 71 163 61 173 51 183)(42 172 72 182 62 192 52 162)(43 191 73 161 63 171 53 181)(44 170 74 180 64 190 54 200)(45 189 75 199 65 169 55 179)(46 168 76 178 66 188 56 198)(47 187 77 197 67 167 57 177)(48 166 78 176 68 186 58 196)(49 185 79 195 69 165 59 175)(50 164 80 174 70 184 60 194)(81 138 91 128 101 158 111 148)(82 157 92 147 102 137 112 127)(83 136 93 126 103 156 113 146)(84 155 94 145 104 135 114 125)(85 134 95 124 105 154 115 144)(86 153 96 143 106 133 116 123)(87 132 97 122 107 152 117 142)(88 151 98 141 108 131 118 121)(89 130 99 160 109 150 119 140)(90 149 100 139 110 129 120 159)

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

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

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

72 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B6A6B8A8B8C8D8E8F8G8H10A···10F12A12B12C15A15B20A···20H24A24B24C24D30A···30F40A···40P60A···60H
order122344455668888888810···10121212151520···202424242430···3040···4060···60
size112211222246666202060602···2224442···2202020204···46···64···4

72 irreducible representations

dim1111122222222222222444444
type++++++-++-+-+-+-+-
imageC1C2C2C2C4S3D4Q8D5D6Dic5D10C4×S3C3⋊D4Dic6C8.C4D20Dic10C40.6C4S3×D5C12.53D4S3×Dic5C3⋊D20C15⋊Q8C12.59D20
kernelC12.59D20C3×C4.Dic5C10×C3⋊C8C60.7C4C5×C3⋊C8C4.Dic5C60C2×C30C2×C3⋊C8C2×C20C3⋊C8C2×C12C20C20C2×C10C15C12C2×C6C3C2×C4C5C4C4C22C1
# reps11114111214222244416222228

Matrix representation of C12.59D20 in GL6(𝔽241)

24000000
02400000
0064000
0006400
00002401
00002400
,
85410000
2001220000
00211000
0022923300
000020237
000023939
,
442250000
1061970000
001485000
00339300
000039204
00002202

G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,240,240,0,0,0,0,1,0],[85,200,0,0,0,0,41,122,0,0,0,0,0,0,211,229,0,0,0,0,0,233,0,0,0,0,0,0,202,239,0,0,0,0,37,39],[44,106,0,0,0,0,225,197,0,0,0,0,0,0,148,33,0,0,0,0,50,93,0,0,0,0,0,0,39,2,0,0,0,0,204,202] >;

C12.59D20 in GAP, Magma, Sage, TeX

C_{12}._{59}D_{20}
% in TeX

G:=Group("C12.59D20");
// GroupNames label

G:=SmallGroup(480,69);
// by ID

G=gap.SmallGroup(480,69);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^3,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=b^19>;
// generators/relations

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