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## G = Dic15⋊5C4order 240 = 24·3·5

### 3rd semidirect product of Dic15 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — Dic15⋊5C4
 Chief series C1 — C5 — C15 — C30 — C2×C30 — C6×Dic5 — Dic15⋊5C4
 Lower central C15 — C30 — Dic15⋊5C4
 Upper central C1 — C22

Generators and relations for Dic155C4
G = < a,b,c | a30=c4=1, b2=a15, bab-1=a-1, cac-1=a11, cbc-1=a15b >

Smallest permutation representation of Dic155C4
Regular action 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)
(1 191 16 206)(2 190 17 205)(3 189 18 204)(4 188 19 203)(5 187 20 202)(6 186 21 201)(7 185 22 200)(8 184 23 199)(9 183 24 198)(10 182 25 197)(11 181 26 196)(12 210 27 195)(13 209 28 194)(14 208 29 193)(15 207 30 192)(31 229 46 214)(32 228 47 213)(33 227 48 212)(34 226 49 211)(35 225 50 240)(36 224 51 239)(37 223 52 238)(38 222 53 237)(39 221 54 236)(40 220 55 235)(41 219 56 234)(42 218 57 233)(43 217 58 232)(44 216 59 231)(45 215 60 230)(61 170 76 155)(62 169 77 154)(63 168 78 153)(64 167 79 152)(65 166 80 151)(66 165 81 180)(67 164 82 179)(68 163 83 178)(69 162 84 177)(70 161 85 176)(71 160 86 175)(72 159 87 174)(73 158 88 173)(74 157 89 172)(75 156 90 171)(91 149 106 134)(92 148 107 133)(93 147 108 132)(94 146 109 131)(95 145 110 130)(96 144 111 129)(97 143 112 128)(98 142 113 127)(99 141 114 126)(100 140 115 125)(101 139 116 124)(102 138 117 123)(103 137 118 122)(104 136 119 121)(105 135 120 150)
(1 119 44 75)(2 100 45 86)(3 111 46 67)(4 92 47 78)(5 103 48 89)(6 114 49 70)(7 95 50 81)(8 106 51 62)(9 117 52 73)(10 98 53 84)(11 109 54 65)(12 120 55 76)(13 101 56 87)(14 112 57 68)(15 93 58 79)(16 104 59 90)(17 115 60 71)(18 96 31 82)(19 107 32 63)(20 118 33 74)(21 99 34 85)(22 110 35 66)(23 91 36 77)(24 102 37 88)(25 113 38 69)(26 94 39 80)(27 105 40 61)(28 116 41 72)(29 97 42 83)(30 108 43 64)(121 231 156 206)(122 212 157 187)(123 223 158 198)(124 234 159 209)(125 215 160 190)(126 226 161 201)(127 237 162 182)(128 218 163 193)(129 229 164 204)(130 240 165 185)(131 221 166 196)(132 232 167 207)(133 213 168 188)(134 224 169 199)(135 235 170 210)(136 216 171 191)(137 227 172 202)(138 238 173 183)(139 219 174 194)(140 230 175 205)(141 211 176 186)(142 222 177 197)(143 233 178 208)(144 214 179 189)(145 225 180 200)(146 236 151 181)(147 217 152 192)(148 228 153 203)(149 239 154 184)(150 220 155 195)

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + + + - - + + - - image C1 C2 C2 C2 C4 S3 D4 Q8 D5 D6 D10 Dic6 C4×S3 C3⋊D4 Dic10 C4×D5 C5⋊D4 S3×D5 D30.C2 C15⋊D4 C15⋊Q8 kernel Dic15⋊5C4 C6×Dic5 C10×Dic3 C2×Dic15 Dic15 C2×Dic5 C30 C30 C2×Dic3 C2×C10 C2×C6 C10 C10 C10 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 1 2 1 2 2 2 2 4 4 4 2 2 2 2

Matrix representation of Dic155C4 in GL4(𝔽61) generated by

 60 1 0 0 16 44 0 0 0 0 0 60 0 0 1 1
,
 0 18 0 0 17 0 0 0 0 0 9 18 0 0 9 52
,
 50 0 0 0 0 50 0 0 0 0 60 0 0 0 1 1
G:=sub<GL(4,GF(61))| [60,16,0,0,1,44,0,0,0,0,0,1,0,0,60,1],[0,17,0,0,18,0,0,0,0,0,9,9,0,0,18,52],[50,0,0,0,0,50,0,0,0,0,60,1,0,0,0,1] >;

Dic155C4 in GAP, Magma, Sage, TeX

{\rm Dic}_{15}\rtimes_5C_4
% in TeX

G:=Group("Dic15:5C4");
// GroupNames label

G:=SmallGroup(240,30);
// by ID

G=gap.SmallGroup(240,30);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,31,490,6917]);
// Polycyclic

G:=Group<a,b,c|a^30=c^4=1,b^2=a^15,b*a*b^-1=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations

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