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## G = C15×SD32order 480 = 25·3·5

### Direct product of C15 and SD32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C15×SD32
 Chief series C1 — C2 — C4 — C8 — C40 — C120 — C15×Q16 — C15×SD32
 Lower central C1 — C2 — C4 — C8 — C15×SD32
 Upper central C1 — C30 — C60 — C120 — C15×SD32

Generators and relations for C15×SD32
G = < a,b,c | a15=b16=c2=1, ab=ba, ac=ca, cbc=b7 >

Smallest permutation representation of C15×SD32
On 240 points
Generators in S240
(1 200 224 23 55 45 179 175 105 239 119 154 136 84 65)(2 201 209 24 56 46 180 176 106 240 120 155 137 85 66)(3 202 210 25 57 47 181 161 107 225 121 156 138 86 67)(4 203 211 26 58 48 182 162 108 226 122 157 139 87 68)(5 204 212 27 59 33 183 163 109 227 123 158 140 88 69)(6 205 213 28 60 34 184 164 110 228 124 159 141 89 70)(7 206 214 29 61 35 185 165 111 229 125 160 142 90 71)(8 207 215 30 62 36 186 166 112 230 126 145 143 91 72)(9 208 216 31 63 37 187 167 97 231 127 146 144 92 73)(10 193 217 32 64 38 188 168 98 232 128 147 129 93 74)(11 194 218 17 49 39 189 169 99 233 113 148 130 94 75)(12 195 219 18 50 40 190 170 100 234 114 149 131 95 76)(13 196 220 19 51 41 191 171 101 235 115 150 132 96 77)(14 197 221 20 52 42 192 172 102 236 116 151 133 81 78)(15 198 222 21 53 43 177 173 103 237 117 152 134 82 79)(16 199 223 22 54 44 178 174 104 238 118 153 135 83 80)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 29)(18 20)(19 27)(21 25)(22 32)(24 30)(26 28)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(66 72)(67 79)(68 70)(69 77)(71 75)(74 80)(76 78)(81 95)(82 86)(83 93)(85 91)(87 89)(88 96)(90 94)(98 104)(99 111)(100 102)(101 109)(103 107)(106 112)(108 110)(113 125)(114 116)(115 123)(117 121)(118 128)(120 126)(122 124)(129 135)(130 142)(131 133)(132 140)(134 138)(137 143)(139 141)(145 155)(147 153)(148 160)(149 151)(150 158)(152 156)(157 159)(161 173)(162 164)(163 171)(165 169)(166 176)(168 174)(170 172)(177 181)(178 188)(180 186)(182 184)(183 191)(185 189)(190 192)(193 199)(194 206)(195 197)(196 204)(198 202)(201 207)(203 205)(209 215)(210 222)(211 213)(212 220)(214 218)(217 223)(219 221)(225 237)(226 228)(227 235)(229 233)(230 240)(232 238)(234 236)

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

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

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

165 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 15A ··· 15H 16A 16B 16C 16D 20A 20B 20C 20D 20E 20F 20G 20H 24A 24B 24C 24D 30A ··· 30H 30I ··· 30P 40A ··· 40H 48A ··· 48H 60A ··· 60H 60I ··· 60P 80A ··· 80P 120A ··· 120P 240A ··· 240AF order 1 2 2 3 3 4 4 5 5 5 5 6 6 6 6 8 8 10 10 10 10 10 10 10 10 12 12 12 12 15 ··· 15 16 16 16 16 20 20 20 20 20 20 20 20 24 24 24 24 30 ··· 30 30 ··· 30 40 ··· 40 48 ··· 48 60 ··· 60 60 ··· 60 80 ··· 80 120 ··· 120 240 ··· 240 size 1 1 8 1 1 2 8 1 1 1 1 1 1 8 8 2 2 1 1 1 1 8 8 8 8 2 2 8 8 1 ··· 1 2 2 2 2 2 2 2 2 8 8 8 8 2 2 2 2 1 ··· 1 8 ··· 8 2 ··· 2 2 ··· 2 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2 2 ··· 2

165 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 D8 C3×D4 SD32 C5×D4 C3×D8 C5×D8 C3×SD32 D4×C15 C5×SD32 C15×D8 C15×SD32 kernel C15×SD32 C240 C15×D8 C15×Q16 C5×SD32 C3×SD32 C80 C5×D8 C5×Q16 C48 C3×D8 C3×Q16 SD32 C16 D8 Q16 C60 C30 C20 C15 C12 C10 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 4 4 4 8 8 8 8 1 2 2 4 4 4 8 8 8 16 16 32

Matrix representation of C15×SD32 in GL3(𝔽241) generated by

 225 0 0 0 91 0 0 0 91
,
 240 0 0 0 103 41 0 200 103
,
 240 0 0 0 1 0 0 0 240
G:=sub<GL(3,GF(241))| [225,0,0,0,91,0,0,0,91],[240,0,0,0,103,200,0,41,103],[240,0,0,0,1,0,0,0,240] >;

C15×SD32 in GAP, Magma, Sage, TeX

C_{15}\times {\rm SD}_{32}
% in TeX

G:=Group("C15xSD32");
// GroupNames label

G:=SmallGroup(480,215);
// by ID

G=gap.SmallGroup(480,215);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,1680,869,6304,3161,242,15125,7572,124]);
// Polycyclic

G:=Group<a,b,c|a^15=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations

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