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## G = C4○D4×C30order 480 = 25·3·5

### Direct product of C30 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C30
 Chief series C1 — C2 — C10 — C30 — C2×C30 — D4×C15 — C15×C4○D4 — C4○D4×C30
 Lower central C1 — C2 — C4○D4×C30
 Upper central C1 — C2×C60 — C4○D4×C30

Generators and relations for C4○D4×C30
G = < a,b,c,d | a30=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 376 in 328 conjugacy classes, 280 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C30, C30, C30, C2×C4○D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C60, C2×C30, C2×C30, C2×C30, C22×C20, D4×C10, Q8×C10, C5×C4○D4, C6×C4○D4, C2×C60, C2×C60, D4×C15, Q8×C15, C22×C30, C10×C4○D4, C22×C60, D4×C30, Q8×C30, C15×C4○D4, C4○D4×C30
Quotients: C1, C2, C3, C22, C5, C6, C23, C10, C2×C6, C15, C4○D4, C24, C2×C10, C22×C6, C30, C2×C4○D4, C22×C10, C3×C4○D4, C23×C6, C2×C30, C5×C4○D4, C23×C10, C6×C4○D4, C22×C30, C10×C4○D4, C15×C4○D4, C23×C30, C4○D4×C30

Smallest permutation representation of C4○D4×C30
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 34 71 91)(2 35 72 92)(3 36 73 93)(4 37 74 94)(5 38 75 95)(6 39 76 96)(7 40 77 97)(8 41 78 98)(9 42 79 99)(10 43 80 100)(11 44 81 101)(12 45 82 102)(13 46 83 103)(14 47 84 104)(15 48 85 105)(16 49 86 106)(17 50 87 107)(18 51 88 108)(19 52 89 109)(20 53 90 110)(21 54 61 111)(22 55 62 112)(23 56 63 113)(24 57 64 114)(25 58 65 115)(26 59 66 116)(27 60 67 117)(28 31 68 118)(29 32 69 119)(30 33 70 120)(121 226 158 198)(122 227 159 199)(123 228 160 200)(124 229 161 201)(125 230 162 202)(126 231 163 203)(127 232 164 204)(128 233 165 205)(129 234 166 206)(130 235 167 207)(131 236 168 208)(132 237 169 209)(133 238 170 210)(134 239 171 181)(135 240 172 182)(136 211 173 183)(137 212 174 184)(138 213 175 185)(139 214 176 186)(140 215 177 187)(141 216 178 188)(142 217 179 189)(143 218 180 190)(144 219 151 191)(145 220 152 192)(146 221 153 193)(147 222 154 194)(148 223 155 195)(149 224 156 196)(150 225 157 197)
(1 175 71 138)(2 176 72 139)(3 177 73 140)(4 178 74 141)(5 179 75 142)(6 180 76 143)(7 151 77 144)(8 152 78 145)(9 153 79 146)(10 154 80 147)(11 155 81 148)(12 156 82 149)(13 157 83 150)(14 158 84 121)(15 159 85 122)(16 160 86 123)(17 161 87 124)(18 162 88 125)(19 163 89 126)(20 164 90 127)(21 165 61 128)(22 166 62 129)(23 167 63 130)(24 168 64 131)(25 169 65 132)(26 170 66 133)(27 171 67 134)(28 172 68 135)(29 173 69 136)(30 174 70 137)(31 182 118 240)(32 183 119 211)(33 184 120 212)(34 185 91 213)(35 186 92 214)(36 187 93 215)(37 188 94 216)(38 189 95 217)(39 190 96 218)(40 191 97 219)(41 192 98 220)(42 193 99 221)(43 194 100 222)(44 195 101 223)(45 196 102 224)(46 197 103 225)(47 198 104 226)(48 199 105 227)(49 200 106 228)(50 201 107 229)(51 202 108 230)(52 203 109 231)(53 204 110 232)(54 205 111 233)(55 206 112 234)(56 207 113 235)(57 208 114 236)(58 209 115 237)(59 210 116 238)(60 181 117 239)
(1 86)(2 87)(3 88)(4 89)(5 90)(6 61)(7 62)(8 63)(9 64)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 72)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 109)(38 110)(39 111)(40 112)(41 113)(42 114)(43 115)(44 116)(45 117)(46 118)(47 119)(48 120)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 97)(56 98)(57 99)(58 100)(59 101)(60 102)(121 136)(122 137)(123 138)(124 139)(125 140)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(151 166)(152 167)(153 168)(154 169)(155 170)(156 171)(157 172)(158 173)(159 174)(160 175)(161 176)(162 177)(163 178)(164 179)(165 180)(181 196)(182 197)(183 198)(184 199)(185 200)(186 201)(187 202)(188 203)(189 204)(190 205)(191 206)(192 207)(193 208)(194 209)(195 210)(211 226)(212 227)(213 228)(214 229)(215 230)(216 231)(217 232)(218 233)(219 234)(220 235)(221 236)(222 237)(223 238)(224 239)(225 240)

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

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

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

300 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A 4B 4C 4D 4E ··· 4J 5A 5B 5C 5D 6A ··· 6F 6G ··· 6R 10A ··· 10L 10M ··· 10AJ 12A ··· 12H 12I ··· 12T 15A ··· 15H 20A ··· 20P 20Q ··· 20AN 30A ··· 30X 30Y ··· 30BT 60A ··· 60AF 60AG ··· 60CB order 1 2 2 2 2 ··· 2 3 3 4 4 4 4 4 ··· 4 5 5 5 5 6 ··· 6 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 12 ··· 12 15 ··· 15 20 ··· 20 20 ··· 20 30 ··· 30 30 ··· 30 60 ··· 60 60 ··· 60 size 1 1 1 1 2 ··· 2 1 1 1 1 1 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

300 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C5 C6 C6 C6 C6 C10 C10 C10 C10 C15 C30 C30 C30 C30 C4○D4 C3×C4○D4 C5×C4○D4 C15×C4○D4 kernel C4○D4×C30 C22×C60 D4×C30 Q8×C30 C15×C4○D4 C10×C4○D4 C6×C4○D4 C22×C20 D4×C10 Q8×C10 C5×C4○D4 C22×C12 C6×D4 C6×Q8 C3×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C30 C10 C6 C2 # reps 1 3 3 1 8 2 4 6 6 2 16 12 12 4 32 8 24 24 8 64 4 8 16 32

Matrix representation of C4○D4×C30 in GL4(𝔽61) generated by

 60 0 0 0 0 13 0 0 0 0 34 0 0 0 0 34
,
 1 0 0 0 0 60 0 0 0 0 50 0 0 0 0 50
,
 1 0 0 0 0 60 0 0 0 0 60 2 0 0 60 1
,
 1 0 0 0 0 60 0 0 0 0 60 0 0 0 60 1
G:=sub<GL(4,GF(61))| [60,0,0,0,0,13,0,0,0,0,34,0,0,0,0,34],[1,0,0,0,0,60,0,0,0,0,50,0,0,0,0,50],[1,0,0,0,0,60,0,0,0,0,60,60,0,0,2,1],[1,0,0,0,0,60,0,0,0,0,60,60,0,0,0,1] >;

C4○D4×C30 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{30}
% in TeX

G:=Group("C4oD4xC30");
// GroupNames label

G:=SmallGroup(480,1183);
// by ID

G=gap.SmallGroup(480,1183);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-5,-2,3389,1276]);
// Polycyclic

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

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