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## G = D12⋊Dic5order 480 = 25·3·5

### 3rd semidirect product of D12 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — D12⋊Dic5
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C2×C60 — C3×C4⋊Dic5 — D12⋊Dic5
 Lower central C15 — C30 — C60 — D12⋊Dic5
 Upper central C1 — C22 — C2×C4

Generators and relations for D12⋊Dic5
G = < a,b,c | a60=b4=1, c2=a15, bab-1=a19, cac-1=a29, cbc-1=a15b-1 >

Subgroups: 428 in 100 conjugacy classes, 42 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C10 [×3], C10 [×2], C12 [×2], C12, D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20 [×2], C2×C10, C2×C10 [×4], C3⋊C8, D12 [×2], D12, C2×C12, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], D4⋊C4, C52C8, C2×Dic5, C2×C20, C5×D4 [×3], C22×C10, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C3×Dic5, C60 [×2], S3×C10 [×4], C2×C30, C2×C52C8, C4⋊Dic5, D4×C10, C6.D8, C153C8, C6×Dic5, C5×D12 [×2], C5×D12, C2×C60, S3×C2×C10, D4⋊Dic5, C3×C4⋊Dic5, C2×C153C8, C10×D12, D12⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D5, D6, C22⋊C4, D8, SD16, Dic5 [×2], D10, C4×S3, D12, C3⋊D4, D4⋊C4, C2×Dic5, C5⋊D4 [×2], D6⋊C4, D4⋊S3, Q82S3, S3×D5, D4⋊D5, D4.D5, C23.D5, C6.D8, S3×Dic5, C15⋊D4, C5⋊D12, D4⋊Dic5, C15⋊D8, C20.D6, D6⋊Dic5, D12⋊Dic5

Smallest permutation representation of D12⋊Dic5
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 97 131 216)(2 116 132 235)(3 75 133 194)(4 94 134 213)(5 113 135 232)(6 72 136 191)(7 91 137 210)(8 110 138 229)(9 69 139 188)(10 88 140 207)(11 107 141 226)(12 66 142 185)(13 85 143 204)(14 104 144 223)(15 63 145 182)(16 82 146 201)(17 101 147 220)(18 120 148 239)(19 79 149 198)(20 98 150 217)(21 117 151 236)(22 76 152 195)(23 95 153 214)(24 114 154 233)(25 73 155 192)(26 92 156 211)(27 111 157 230)(28 70 158 189)(29 89 159 208)(30 108 160 227)(31 67 161 186)(32 86 162 205)(33 105 163 224)(34 64 164 183)(35 83 165 202)(36 102 166 221)(37 61 167 240)(38 80 168 199)(39 99 169 218)(40 118 170 237)(41 77 171 196)(42 96 172 215)(43 115 173 234)(44 74 174 193)(45 93 175 212)(46 112 176 231)(47 71 177 190)(48 90 178 209)(49 109 179 228)(50 68 180 187)(51 87 121 206)(52 106 122 225)(53 65 123 184)(54 84 124 203)(55 103 125 222)(56 62 126 181)(57 81 127 200)(58 100 128 219)(59 119 129 238)(60 78 130 197)
(1 112 16 67 31 82 46 97)(2 81 17 96 32 111 47 66)(3 110 18 65 33 80 48 95)(4 79 19 94 34 109 49 64)(5 108 20 63 35 78 50 93)(6 77 21 92 36 107 51 62)(7 106 22 61 37 76 52 91)(8 75 23 90 38 105 53 120)(9 104 24 119 39 74 54 89)(10 73 25 88 40 103 55 118)(11 102 26 117 41 72 56 87)(12 71 27 86 42 101 57 116)(13 100 28 115 43 70 58 85)(14 69 29 84 44 99 59 114)(15 98 30 113 45 68 60 83)(121 181 136 196 151 211 166 226)(122 210 137 225 152 240 167 195)(123 239 138 194 153 209 168 224)(124 208 139 223 154 238 169 193)(125 237 140 192 155 207 170 222)(126 206 141 221 156 236 171 191)(127 235 142 190 157 205 172 220)(128 204 143 219 158 234 173 189)(129 233 144 188 159 203 174 218)(130 202 145 217 160 232 175 187)(131 231 146 186 161 201 176 216)(132 200 147 215 162 230 177 185)(133 229 148 184 163 199 178 214)(134 198 149 213 164 228 179 183)(135 227 150 182 165 197 180 212)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 8A 8B 8C 8D 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 12 12 2 2 2 20 20 2 2 2 2 2 30 30 30 30 2 ··· 2 12 ··· 12 4 4 20 20 20 20 4 4 4 4 4 4 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + - + + + + + + - - + - image C1 C2 C2 C2 C4 S3 D4 D4 D5 D6 D8 SD16 Dic5 D10 C4×S3 D12 C3⋊D4 C5⋊D4 C5⋊D4 D4⋊S3 Q8⋊2S3 S3×D5 D4⋊D5 D4.D5 S3×Dic5 C5⋊D12 C15⋊D4 C15⋊D8 C20.D6 kernel D12⋊Dic5 C3×C4⋊Dic5 C2×C15⋊3C8 C10×D12 C5×D12 C4⋊Dic5 C60 C2×C30 C2×D12 C2×C20 C30 C30 D12 C2×C12 C20 C20 C2×C10 C12 C2×C6 C10 C10 C2×C4 C6 C6 C4 C4 C22 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 2 2 4 2 2 2 2 4 4 1 1 2 2 2 2 2 2 4 4

Matrix representation of D12⋊Dic5 in GL6(𝔽241)

 2 49 0 0 0 0 177 240 0 0 0 0 0 0 1 51 0 0 0 0 190 51 0 0 0 0 0 0 240 164 0 0 0 0 144 1
,
 177 0 0 0 0 0 0 177 0 0 0 0 0 0 91 166 0 0 0 0 104 150 0 0 0 0 0 0 203 224 0 0 0 0 85 38
,
 177 238 0 0 0 0 0 64 0 0 0 0 0 0 150 75 0 0 0 0 137 91 0 0 0 0 0 0 0 224 0 0 0 0 85 38

G:=sub<GL(6,GF(241))| [2,177,0,0,0,0,49,240,0,0,0,0,0,0,1,190,0,0,0,0,51,51,0,0,0,0,0,0,240,144,0,0,0,0,164,1],[177,0,0,0,0,0,0,177,0,0,0,0,0,0,91,104,0,0,0,0,166,150,0,0,0,0,0,0,203,85,0,0,0,0,224,38],[177,0,0,0,0,0,238,64,0,0,0,0,0,0,150,137,0,0,0,0,75,91,0,0,0,0,0,0,0,85,0,0,0,0,224,38] >;

D12⋊Dic5 in GAP, Magma, Sage, TeX

D_{12}\rtimes {\rm Dic}_5
% in TeX

G:=Group("D12:Dic5");
// GroupNames label

G:=SmallGroup(480,42);
// by ID

G=gap.SmallGroup(480,42);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,675,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c|a^60=b^4=1,c^2=a^15,b*a*b^-1=a^19,c*a*c^-1=a^29,c*b*c^-1=a^15*b^-1>;
// generators/relations

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