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G = C12.6D20order 480 = 25·3·5

6th non-split extension by C12 of D20 acting via D20/C10=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.6D20, C60.30D4, (C2×C20).43D6, (C2×C12).44D10, C12.8(C5⋊D4), (C6×Dic5).1C4, C4.Dic3.2D5, C60.7C4.3C2, C20.78(C3⋊D4), C154(C4.10D4), C4.20(C3⋊D20), C33(C4.12D20), C4.20(C15⋊D4), C52(C12.10D4), (C2×C60).30C22, (C6×Dic10).1C2, (C2×Dic10).6S3, C22.4(D5×Dic3), C30.44(C22⋊C4), (C2×Dic5).1Dic3, C6.25(D10⋊C4), C2.4(D10⋊Dic3), C10.14(C6.D4), (C2×C4).4(S3×D5), (C2×C6).46(C4×D5), (C2×C30).83(C2×C4), (C5×C4.Dic3).1C2, (C2×C10).22(C2×Dic3), SmallGroup(480,37)

Series: Derived Chief Lower central Upper central

C1C2×C30 — C12.6D20
C1C5C15C30C60C2×C60C6×Dic10 — C12.6D20
C15C30C2×C30 — C12.6D20
C1C2C2×C4

Generators and relations for C12.6D20
 G = < a,b,c | a12=1, b20=a6, c2=a3, bab-1=a-1, cac-1=a5, cbc-1=a3b19 >

Subgroups: 284 in 76 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C5, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C10, C10, C12 [×2], C12 [×2], C2×C6, C15, M4(2) [×2], C2×Q8, Dic5 [×2], C20 [×2], C2×C10, C3⋊C8 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C30, C30, C4.10D4, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×C20, C4.Dic3, C4.Dic3, C6×Q8, C3×Dic5 [×2], C60 [×2], C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C12.10D4, C5×C3⋊C8, C153C8, C3×Dic10 [×2], C6×Dic5 [×2], C2×C60, C4.12D20, C5×C4.Dic3, C60.7C4, C6×Dic10, C12.6D20
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, Dic3 [×2], D6, C22⋊C4, D10, C2×Dic3, C3⋊D4 [×2], C4.10D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C12.10D4, D5×Dic3, C15⋊D4, C3⋊D20, C4.12D20, D10⋊Dic3, C12.6D20

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

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

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

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

57 conjugacy classes

class 1 2A2B 3 4A4B4C4D5A5B6A6B6C8A8B8C8D10A10B10C10D12A12B12C12D12E12F15A15B20A20B20C20D20E20F30A···30F40A···40H60A···60H
order1223444455666888810101010121212121212151520202020202030···3040···4060···60
size1122222020222221212606022444420202020442222444···412···124···4

57 irreducible representations

dim11111222222222244444444
type+++++++-+++-+-+--
imageC1C2C2C2C4S3D4D5Dic3D6D10C3⋊D4D20C5⋊D4C4×D5C4.10D4S3×D5C12.10D4C15⋊D4C3⋊D20D5×Dic3C4.12D20C12.6D20
kernelC12.6D20C5×C4.Dic3C60.7C4C6×Dic10C6×Dic5C2×Dic10C60C4.Dic3C2×Dic5C2×C20C2×C12C20C12C12C2×C6C15C2×C4C5C4C4C22C3C1
# reps11114122212444412222248

Matrix representation of C12.6D20 in GL6(𝔽241)

22600000
0160000
0019723800
0034400
0000443
0000238197
,
01080000
21200000
000001
000024051
0034400
001977800
,
01080000
2900000
00003566
0000164206
0010414800
0015013700

G:=sub<GL(6,GF(241))| [226,0,0,0,0,0,0,16,0,0,0,0,0,0,197,3,0,0,0,0,238,44,0,0,0,0,0,0,44,238,0,0,0,0,3,197],[0,212,0,0,0,0,108,0,0,0,0,0,0,0,0,0,3,197,0,0,0,0,44,78,0,0,0,240,0,0,0,0,1,51,0,0],[0,29,0,0,0,0,108,0,0,0,0,0,0,0,0,0,104,150,0,0,0,0,148,137,0,0,35,164,0,0,0,0,66,206,0,0] >;

C12.6D20 in GAP, Magma, Sage, TeX

C_{12}._6D_{20}
% in TeX

G:=Group("C12.6D20");
// GroupNames label

G:=SmallGroup(480,37);
// by ID

G=gap.SmallGroup(480,37);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,422,100,346,1356,18822]);
// Polycyclic

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

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