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## G = C5⋊Dic12order 240 = 24·3·5

### The semidirect product of C5 and Dic12 acting via Dic12/Dic6=C2

Aliases: C153Q16, C20.8D6, C52Dic12, C30.12D4, C10.9D12, C12.25D10, Dic6.1D5, C60.18C22, Dic30.4C2, C52C8.1S3, C4.11(S3×D5), C31(C5⋊Q16), C6.4(C5⋊D4), C2.7(C5⋊D12), (C5×Dic6).1C2, (C3×C52C8).1C2, SmallGroup(240,24)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C5⋊Dic12
 Chief series C1 — C5 — C15 — C30 — C60 — C3×C5⋊2C8 — C5⋊Dic12
 Lower central C15 — C30 — C60 — C5⋊Dic12
 Upper central C1 — C2 — C4

Generators and relations for C5⋊Dic12
G = < a,b,c | a5=b24=1, c2=b12, bab-1=a-1, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C5⋊Dic12
Regular action on 240 points
Generators in S240
(1 197 236 186 154)(2 155 187 237 198)(3 199 238 188 156)(4 157 189 239 200)(5 201 240 190 158)(6 159 191 217 202)(7 203 218 192 160)(8 161 169 219 204)(9 205 220 170 162)(10 163 171 221 206)(11 207 222 172 164)(12 165 173 223 208)(13 209 224 174 166)(14 167 175 225 210)(15 211 226 176 168)(16 145 177 227 212)(17 213 228 178 146)(18 147 179 229 214)(19 215 230 180 148)(20 149 181 231 216)(21 193 232 182 150)(22 151 183 233 194)(23 195 234 184 152)(24 153 185 235 196)(25 76 67 142 118)(26 119 143 68 77)(27 78 69 144 120)(28 97 121 70 79)(29 80 71 122 98)(30 99 123 72 81)(31 82 49 124 100)(32 101 125 50 83)(33 84 51 126 102)(34 103 127 52 85)(35 86 53 128 104)(36 105 129 54 87)(37 88 55 130 106)(38 107 131 56 89)(39 90 57 132 108)(40 109 133 58 91)(41 92 59 134 110)(42 111 135 60 93)(43 94 61 136 112)(44 113 137 62 95)(45 96 63 138 114)(46 115 139 64 73)(47 74 65 140 116)(48 117 141 66 75)
(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 92 13 80)(2 91 14 79)(3 90 15 78)(4 89 16 77)(5 88 17 76)(6 87 18 75)(7 86 19 74)(8 85 20 73)(9 84 21 96)(10 83 22 95)(11 82 23 94)(12 81 24 93)(25 158 37 146)(26 157 38 145)(27 156 39 168)(28 155 40 167)(29 154 41 166)(30 153 42 165)(31 152 43 164)(32 151 44 163)(33 150 45 162)(34 149 46 161)(35 148 47 160)(36 147 48 159)(49 195 61 207)(50 194 62 206)(51 193 63 205)(52 216 64 204)(53 215 65 203)(54 214 66 202)(55 213 67 201)(56 212 68 200)(57 211 69 199)(58 210 70 198)(59 209 71 197)(60 208 72 196)(97 187 109 175)(98 186 110 174)(99 185 111 173)(100 184 112 172)(101 183 113 171)(102 182 114 170)(103 181 115 169)(104 180 116 192)(105 179 117 191)(106 178 118 190)(107 177 119 189)(108 176 120 188)(121 237 133 225)(122 236 134 224)(123 235 135 223)(124 234 136 222)(125 233 137 221)(126 232 138 220)(127 231 139 219)(128 230 140 218)(129 229 141 217)(130 228 142 240)(131 227 143 239)(132 226 144 238)

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

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

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

C5⋊Dic12 is a maximal subgroup of
D5×Dic12  Dic60⋊C2  C24.2D10  C40.31D6  C20.60D12  C20.D12  D12.33D10  C60.10C23  D30.9D4  D20.24D6  D20.10D6  S3×C5⋊Q16  D15⋊Q16  D20.28D6  D20.17D6
C5⋊Dic12 is a maximal quotient of
C10.Dic12  Dic3015C4  C60.8Q8

33 conjugacy classes

 class 1 2 3 4A 4B 4C 5A 5B 6 8A 8B 10A 10B 12A 12B 15A 15B 20A 20B 20C 20D 20E 20F 24A 24B 24C 24D 30A 30B 60A 60B 60C 60D order 1 2 3 4 4 4 5 5 6 8 8 10 10 12 12 15 15 20 20 20 20 20 20 24 24 24 24 30 30 60 60 60 60 size 1 1 2 2 12 60 2 2 2 10 10 2 2 2 2 4 4 4 4 12 12 12 12 10 10 10 10 4 4 4 4 4 4

33 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + - + - + - image C1 C2 C2 C2 S3 D4 D5 D6 Q16 D10 D12 C5⋊D4 Dic12 S3×D5 C5⋊Q16 C5⋊D12 C5⋊Dic12 kernel C5⋊Dic12 C3×C5⋊2C8 C5×Dic6 Dic30 C5⋊2C8 C30 Dic6 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 1 1 2 1 2 2 2 4 4 2 2 2 4

Matrix representation of C5⋊Dic12 in GL4(𝔽241) generated by

 1 0 0 0 0 1 0 0 0 0 240 1 0 0 50 190
,
 9 114 0 0 127 136 0 0 0 0 228 42 0 0 237 13
,
 140 70 0 0 171 101 0 0 0 0 240 0 0 0 0 240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[9,127,0,0,114,136,0,0,0,0,228,237,0,0,42,13],[140,171,0,0,70,101,0,0,0,0,240,0,0,0,0,240] >;

C5⋊Dic12 in GAP, Magma, Sage, TeX

C_5\rtimes {\rm Dic}_{12}
% in TeX

G:=Group("C5:Dic12");
// GroupNames label

G:=SmallGroup(240,24);
// by ID

G=gap.SmallGroup(240,24);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,73,55,116,50,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^5=b^24=1,c^2=b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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