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## G = D8×C30order 480 = 25·3·5

### Direct product of C30 and D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — D8×C30
 Chief series C1 — C2 — C4 — C20 — C60 — D4×C15 — C15×D8 — D8×C30
 Lower central C1 — C2 — C4 — D8×C30
 Upper central C1 — C2×C30 — C2×C60 — D8×C30

Generators and relations for D8×C30
G = < a,b,c | a30=b8=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 280 in 152 conjugacy classes, 88 normal (32 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C8, C2×C4, D4, D4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C15, C2×C8, D8, C2×D4, C20, C2×C10, C2×C10, C24, C2×C12, C3×D4, C3×D4, C22×C6, C30, C30, C30, C2×D8, C40, C2×C20, C5×D4, C5×D4, C22×C10, C2×C24, C3×D8, C6×D4, C60, C2×C30, C2×C30, C2×C40, C5×D8, D4×C10, C6×D8, C120, C2×C60, D4×C15, D4×C15, C22×C30, C10×D8, C2×C120, C15×D8, D4×C30, D8×C30
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, D8, C2×D4, C2×C10, C3×D4, C22×C6, C30, C2×D8, C5×D4, C22×C10, C3×D8, C6×D4, C2×C30, C5×D8, D4×C10, C6×D8, D4×C15, C22×C30, C10×D8, C15×D8, D4×C30, D8×C30

Smallest permutation representation of D8×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)
(1 177 217 86 44 103 121 202)(2 178 218 87 45 104 122 203)(3 179 219 88 46 105 123 204)(4 180 220 89 47 106 124 205)(5 151 221 90 48 107 125 206)(6 152 222 61 49 108 126 207)(7 153 223 62 50 109 127 208)(8 154 224 63 51 110 128 209)(9 155 225 64 52 111 129 210)(10 156 226 65 53 112 130 181)(11 157 227 66 54 113 131 182)(12 158 228 67 55 114 132 183)(13 159 229 68 56 115 133 184)(14 160 230 69 57 116 134 185)(15 161 231 70 58 117 135 186)(16 162 232 71 59 118 136 187)(17 163 233 72 60 119 137 188)(18 164 234 73 31 120 138 189)(19 165 235 74 32 91 139 190)(20 166 236 75 33 92 140 191)(21 167 237 76 34 93 141 192)(22 168 238 77 35 94 142 193)(23 169 239 78 36 95 143 194)(24 170 240 79 37 96 144 195)(25 171 211 80 38 97 145 196)(26 172 212 81 39 98 146 197)(27 173 213 82 40 99 147 198)(28 174 214 83 41 100 148 199)(29 175 215 84 42 101 149 200)(30 176 216 85 43 102 150 201)
(1 59)(2 60)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(61 167)(62 168)(63 169)(64 170)(65 171)(66 172)(67 173)(68 174)(69 175)(70 176)(71 177)(72 178)(73 179)(74 180)(75 151)(76 152)(77 153)(78 154)(79 155)(80 156)(81 157)(82 158)(83 159)(84 160)(85 161)(86 162)(87 163)(88 164)(89 165)(90 166)(91 205)(92 206)(93 207)(94 208)(95 209)(96 210)(97 181)(98 182)(99 183)(100 184)(101 185)(102 186)(103 187)(104 188)(105 189)(106 190)(107 191)(108 192)(109 193)(110 194)(111 195)(112 196)(113 197)(114 198)(115 199)(116 200)(117 201)(118 202)(119 203)(120 204)(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)(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), (1,177,217,86,44,103,121,202)(2,178,218,87,45,104,122,203)(3,179,219,88,46,105,123,204)(4,180,220,89,47,106,124,205)(5,151,221,90,48,107,125,206)(6,152,222,61,49,108,126,207)(7,153,223,62,50,109,127,208)(8,154,224,63,51,110,128,209)(9,155,225,64,52,111,129,210)(10,156,226,65,53,112,130,181)(11,157,227,66,54,113,131,182)(12,158,228,67,55,114,132,183)(13,159,229,68,56,115,133,184)(14,160,230,69,57,116,134,185)(15,161,231,70,58,117,135,186)(16,162,232,71,59,118,136,187)(17,163,233,72,60,119,137,188)(18,164,234,73,31,120,138,189)(19,165,235,74,32,91,139,190)(20,166,236,75,33,92,140,191)(21,167,237,76,34,93,141,192)(22,168,238,77,35,94,142,193)(23,169,239,78,36,95,143,194)(24,170,240,79,37,96,144,195)(25,171,211,80,38,97,145,196)(26,172,212,81,39,98,146,197)(27,173,213,82,40,99,147,198)(28,174,214,83,41,100,148,199)(29,175,215,84,42,101,149,200)(30,176,216,85,43,102,150,201), (1,59)(2,60)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,177)(72,178)(73,179)(74,180)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)(81,157)(82,158)(83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(89,165)(90,166)(91,205)(92,206)(93,207)(94,208)(95,209)(96,210)(97,181)(98,182)(99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,190)(107,191)(108,192)(109,193)(110,194)(111,195)(112,196)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204)(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)(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), (1,177,217,86,44,103,121,202)(2,178,218,87,45,104,122,203)(3,179,219,88,46,105,123,204)(4,180,220,89,47,106,124,205)(5,151,221,90,48,107,125,206)(6,152,222,61,49,108,126,207)(7,153,223,62,50,109,127,208)(8,154,224,63,51,110,128,209)(9,155,225,64,52,111,129,210)(10,156,226,65,53,112,130,181)(11,157,227,66,54,113,131,182)(12,158,228,67,55,114,132,183)(13,159,229,68,56,115,133,184)(14,160,230,69,57,116,134,185)(15,161,231,70,58,117,135,186)(16,162,232,71,59,118,136,187)(17,163,233,72,60,119,137,188)(18,164,234,73,31,120,138,189)(19,165,235,74,32,91,139,190)(20,166,236,75,33,92,140,191)(21,167,237,76,34,93,141,192)(22,168,238,77,35,94,142,193)(23,169,239,78,36,95,143,194)(24,170,240,79,37,96,144,195)(25,171,211,80,38,97,145,196)(26,172,212,81,39,98,146,197)(27,173,213,82,40,99,147,198)(28,174,214,83,41,100,148,199)(29,175,215,84,42,101,149,200)(30,176,216,85,43,102,150,201), (1,59)(2,60)(3,31)(4,32)(5,33)(6,34)(7,35)(8,36)(9,37)(10,38)(11,39)(12,40)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(61,167)(62,168)(63,169)(64,170)(65,171)(66,172)(67,173)(68,174)(69,175)(70,176)(71,177)(72,178)(73,179)(74,180)(75,151)(76,152)(77,153)(78,154)(79,155)(80,156)(81,157)(82,158)(83,159)(84,160)(85,161)(86,162)(87,163)(88,164)(89,165)(90,166)(91,205)(92,206)(93,207)(94,208)(95,209)(96,210)(97,181)(98,182)(99,183)(100,184)(101,185)(102,186)(103,187)(104,188)(105,189)(106,190)(107,191)(108,192)(109,193)(110,194)(111,195)(112,196)(113,197)(114,198)(115,199)(116,200)(117,201)(118,202)(119,203)(120,204)(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)(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)], [(1,177,217,86,44,103,121,202),(2,178,218,87,45,104,122,203),(3,179,219,88,46,105,123,204),(4,180,220,89,47,106,124,205),(5,151,221,90,48,107,125,206),(6,152,222,61,49,108,126,207),(7,153,223,62,50,109,127,208),(8,154,224,63,51,110,128,209),(9,155,225,64,52,111,129,210),(10,156,226,65,53,112,130,181),(11,157,227,66,54,113,131,182),(12,158,228,67,55,114,132,183),(13,159,229,68,56,115,133,184),(14,160,230,69,57,116,134,185),(15,161,231,70,58,117,135,186),(16,162,232,71,59,118,136,187),(17,163,233,72,60,119,137,188),(18,164,234,73,31,120,138,189),(19,165,235,74,32,91,139,190),(20,166,236,75,33,92,140,191),(21,167,237,76,34,93,141,192),(22,168,238,77,35,94,142,193),(23,169,239,78,36,95,143,194),(24,170,240,79,37,96,144,195),(25,171,211,80,38,97,145,196),(26,172,212,81,39,98,146,197),(27,173,213,82,40,99,147,198),(28,174,214,83,41,100,148,199),(29,175,215,84,42,101,149,200),(30,176,216,85,43,102,150,201)], [(1,59),(2,60),(3,31),(4,32),(5,33),(6,34),(7,35),(8,36),(9,37),(10,38),(11,39),(12,40),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(61,167),(62,168),(63,169),(64,170),(65,171),(66,172),(67,173),(68,174),(69,175),(70,176),(71,177),(72,178),(73,179),(74,180),(75,151),(76,152),(77,153),(78,154),(79,155),(80,156),(81,157),(82,158),(83,159),(84,160),(85,161),(86,162),(87,163),(88,164),(89,165),(90,166),(91,205),(92,206),(93,207),(94,208),(95,209),(96,210),(97,181),(98,182),(99,183),(100,184),(101,185),(102,186),(103,187),(104,188),(105,189),(106,190),(107,191),(108,192),(109,193),(110,194),(111,195),(112,196),(113,197),(114,198),(115,199),(116,200),(117,201),(118,202),(119,203),(120,204),(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),(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)]])

210 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 5A 5B 5C 5D 6A ··· 6F 6G ··· 6N 8A 8B 8C 8D 10A ··· 10L 10M ··· 10AB 12A 12B 12C 12D 15A ··· 15H 20A ··· 20H 24A ··· 24H 30A ··· 30X 30Y ··· 30BD 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 2 2 3 3 4 4 5 5 5 5 6 ··· 6 6 ··· 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 15 ··· 15 20 ··· 20 24 ··· 24 30 ··· 30 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 4 4 4 4 1 1 2 2 1 1 1 1 1 ··· 1 4 ··· 4 2 2 2 2 1 ··· 1 4 ··· 4 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 4 ··· 4 2 ··· 2 2 ··· 2 2 ··· 2

210 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C5 C6 C6 C6 C10 C10 C10 C15 C30 C30 C30 D4 D4 D8 C3×D4 C3×D4 C5×D4 C5×D4 C3×D8 C5×D8 D4×C15 D4×C15 C15×D8 kernel D8×C30 C2×C120 C15×D8 D4×C30 C10×D8 C6×D8 C2×C40 C5×D8 D4×C10 C2×C24 C3×D8 C6×D4 C2×D8 C2×C8 D8 C2×D4 C60 C2×C30 C30 C20 C2×C10 C12 C2×C6 C10 C6 C4 C22 C2 # reps 1 1 4 2 2 4 2 8 4 4 16 8 8 8 32 16 1 1 4 2 2 4 4 8 16 8 8 32

Matrix representation of D8×C30 in GL3(𝔽241) generated by

 16 0 0 0 205 0 0 0 205
,
 1 0 0 0 22 11 0 219 0
,
 1 0 0 0 240 240 0 0 1
G:=sub<GL(3,GF(241))| [16,0,0,0,205,0,0,0,205],[1,0,0,0,22,219,0,11,0],[1,0,0,0,240,0,0,240,1] >;

D8×C30 in GAP, Magma, Sage, TeX

D_8\times C_{30}
% in TeX

G:=Group("D8xC30");
// GroupNames label

G:=SmallGroup(480,937);
// by ID

G=gap.SmallGroup(480,937);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,15125,7572,124]);
// Polycyclic

G:=Group<a,b,c|a^30=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
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