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## G = SD16×C30order 480 = 25·3·5

### Direct product of C30 and SD16

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — SD16×C30
 Chief series C1 — C2 — C4 — C20 — C60 — Q8×C15 — C15×SD16 — SD16×C30
 Lower central C1 — C2 — C4 — SD16×C30
 Upper central C1 — C2×C30 — C2×C60 — SD16×C30

Generators and relations for SD16×C30
G = < a,b,c | a30=b8=c2=1, ab=ba, ac=ca, cbc=b3 >

Subgroups: 216 in 136 conjugacy classes, 88 normal (40 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, C6, C6 [×2], C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×2], D4, Q8 [×2], Q8, C23, C10, C10 [×2], C10 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×4], C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×C6, C30, C30 [×2], C30 [×2], C2×SD16, C40 [×2], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C5×Q8, C22×C10, C2×C24, C3×SD16 [×4], C6×D4, C6×Q8, C60 [×2], C60 [×2], C2×C30, C2×C30 [×4], C2×C40, C5×SD16 [×4], D4×C10, Q8×C10, C6×SD16, C120 [×2], C2×C60, C2×C60, D4×C15 [×2], D4×C15, Q8×C15 [×2], Q8×C15, C22×C30, C10×SD16, C2×C120, C15×SD16 [×4], D4×C30, Q8×C30, SD16×C30
Quotients: C1, C2 [×7], C3, C22 [×7], C5, C6 [×7], D4 [×2], C23, C10 [×7], C2×C6 [×7], C15, SD16 [×2], C2×D4, C2×C10 [×7], C3×D4 [×2], C22×C6, C30 [×7], C2×SD16, C5×D4 [×2], C22×C10, C3×SD16 [×2], C6×D4, C2×C30 [×7], C5×SD16 [×2], D4×C10, C6×SD16, D4×C15 [×2], C22×C30, C10×SD16, C15×SD16 [×2], D4×C30, SD16×C30

Smallest permutation representation of SD16×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)
(1 189 121 93 212 174 89 46)(2 190 122 94 213 175 90 47)(3 191 123 95 214 176 61 48)(4 192 124 96 215 177 62 49)(5 193 125 97 216 178 63 50)(6 194 126 98 217 179 64 51)(7 195 127 99 218 180 65 52)(8 196 128 100 219 151 66 53)(9 197 129 101 220 152 67 54)(10 198 130 102 221 153 68 55)(11 199 131 103 222 154 69 56)(12 200 132 104 223 155 70 57)(13 201 133 105 224 156 71 58)(14 202 134 106 225 157 72 59)(15 203 135 107 226 158 73 60)(16 204 136 108 227 159 74 31)(17 205 137 109 228 160 75 32)(18 206 138 110 229 161 76 33)(19 207 139 111 230 162 77 34)(20 208 140 112 231 163 78 35)(21 209 141 113 232 164 79 36)(22 210 142 114 233 165 80 37)(23 181 143 115 234 166 81 38)(24 182 144 116 235 167 82 39)(25 183 145 117 236 168 83 40)(26 184 146 118 237 169 84 41)(27 185 147 119 238 170 85 42)(28 186 148 120 239 171 86 43)(29 187 149 91 240 172 87 44)(30 188 150 92 211 173 88 45)
(31 159)(32 160)(33 161)(34 162)(35 163)(36 164)(37 165)(38 166)(39 167)(40 168)(41 169)(42 170)(43 171)(44 172)(45 173)(46 174)(47 175)(48 176)(49 177)(50 178)(51 179)(52 180)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 123)(62 124)(63 125)(64 126)(65 127)(66 128)(67 129)(68 130)(69 131)(70 132)(71 133)(72 134)(73 135)(74 136)(75 137)(76 138)(77 139)(78 140)(79 141)(80 142)(81 143)(82 144)(83 145)(84 146)(85 147)(86 148)(87 149)(88 150)(89 121)(90 122)(91 187)(92 188)(93 189)(94 190)(95 191)(96 192)(97 193)(98 194)(99 195)(100 196)(101 197)(102 198)(103 199)(104 200)(105 201)(106 202)(107 203)(108 204)(109 205)(110 206)(111 207)(112 208)(113 209)(114 210)(115 181)(116 182)(117 183)(118 184)(119 185)(120 186)

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

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

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

210 conjugacy classes

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

210 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 2 2 2 2 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 D4 D4 SD16 C3×D4 C3×D4 C5×D4 C5×D4 C3×SD16 C5×SD16 D4×C15 D4×C15 C15×SD16 kernel SD16×C30 C2×C120 C15×SD16 D4×C30 Q8×C30 C10×SD16 C6×SD16 C2×C40 C5×SD16 D4×C10 Q8×C10 C2×C24 C3×SD16 C6×D4 C6×Q8 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C60 C2×C30 C30 C20 C2×C10 C12 C2×C6 C10 C6 C4 C22 C2 # reps 1 1 4 1 1 2 4 2 8 2 2 4 16 4 4 8 8 32 8 8 1 1 4 2 2 4 4 8 16 8 8 32

Matrix representation of SD16×C30 in GL3(𝔽241) generated by

 240 0 0 0 141 0 0 0 141
,
 240 0 0 0 203 222 0 38 0
,
 240 0 0 0 1 1 0 0 240
G:=sub<GL(3,GF(241))| [240,0,0,0,141,0,0,0,141],[240,0,0,0,203,38,0,222,0],[240,0,0,0,1,0,0,1,240] >;

SD16×C30 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\times C_{30}
% in TeX

G:=Group("SD16xC30");
// GroupNames label

G:=SmallGroup(480,938);
// by ID

G=gap.SmallGroup(480,938);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,15125,7572,124]);
// 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^3>;
// generators/relations

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