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## G = C15×M5(2)  order 480 = 25·3·5

### Direct product of C15 and M5(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C15×M5(2)
 Chief series C1 — C2 — C4 — C8 — C40 — C120 — C240 — C15×M5(2)
 Lower central C1 — C2 — C15×M5(2)
 Upper central C1 — C120 — C15×M5(2)

Generators and relations for C15×M5(2)
G = < a,b,c | a15=b16=c2=1, ab=ba, ac=ca, cbc=b9 >

Smallest permutation representation of C15×M5(2)
On 240 points
Generators in S240
(1 46 145 134 49 224 180 207 120 66 25 92 105 237 164)(2 47 146 135 50 209 181 208 121 67 26 93 106 238 165)(3 48 147 136 51 210 182 193 122 68 27 94 107 239 166)(4 33 148 137 52 211 183 194 123 69 28 95 108 240 167)(5 34 149 138 53 212 184 195 124 70 29 96 109 225 168)(6 35 150 139 54 213 185 196 125 71 30 81 110 226 169)(7 36 151 140 55 214 186 197 126 72 31 82 111 227 170)(8 37 152 141 56 215 187 198 127 73 32 83 112 228 171)(9 38 153 142 57 216 188 199 128 74 17 84 97 229 172)(10 39 154 143 58 217 189 200 113 75 18 85 98 230 173)(11 40 155 144 59 218 190 201 114 76 19 86 99 231 174)(12 41 156 129 60 219 191 202 115 77 20 87 100 232 175)(13 42 157 130 61 220 192 203 116 78 21 88 101 233 176)(14 43 158 131 62 221 177 204 117 79 22 89 102 234 161)(15 44 159 132 63 222 178 205 118 80 23 90 103 235 162)(16 45 160 133 64 223 179 206 119 65 24 91 104 236 163)
(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 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)(50 58)(52 60)(54 62)(56 64)(65 73)(67 75)(69 77)(71 79)(81 89)(83 91)(85 93)(87 95)(98 106)(100 108)(102 110)(104 112)(113 121)(115 123)(117 125)(119 127)(129 137)(131 139)(133 141)(135 143)(146 154)(148 156)(150 158)(152 160)(161 169)(163 171)(165 173)(167 175)(177 185)(179 187)(181 189)(183 191)(194 202)(196 204)(198 206)(200 208)(209 217)(211 219)(213 221)(215 223)(226 234)(228 236)(230 238)(232 240)

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

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

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

300 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 5A 5B 5C 5D 6A 6B 6C 6D 8A 8B 8C 8D 8E 8F 10A 10B 10C 10D 10E 10F 10G 10H 12A 12B 12C 12D 12E 12F 15A ··· 15H 16A ··· 16H 20A ··· 20H 20I 20J 20K 20L 24A ··· 24H 24I 24J 24K 24L 30A ··· 30H 30I ··· 30P 40A ··· 40P 40Q ··· 40X 48A ··· 48P 60A ··· 60P 60Q ··· 60X 80A ··· 80AF 120A ··· 120AF 120AG ··· 120AV 240A ··· 240BL order 1 2 2 3 3 4 4 4 5 5 5 5 6 6 6 6 8 8 8 8 8 8 10 10 10 10 10 10 10 10 12 12 12 12 12 12 15 ··· 15 16 ··· 16 20 ··· 20 20 20 20 20 24 ··· 24 24 24 24 24 30 ··· 30 30 ··· 30 40 ··· 40 40 ··· 40 48 ··· 48 60 ··· 60 60 ··· 60 80 ··· 80 120 ··· 120 120 ··· 120 240 ··· 240 size 1 1 2 1 1 1 1 2 1 1 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 2 2 2 2 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 1 ··· 1 2 ··· 2 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 1 1 1 1 1 1 1 1 2 2 2 2 type + + + image C1 C2 C2 C3 C4 C4 C5 C6 C6 C8 C8 C10 C10 C12 C12 C15 C20 C20 C24 C24 C30 C30 C40 C40 C60 C60 C120 C120 M5(2) C3×M5(2) C5×M5(2) C15×M5(2) kernel C15×M5(2) C240 C2×C120 C5×M5(2) C120 C2×C60 C3×M5(2) C80 C2×C40 C60 C2×C30 C48 C2×C24 C40 C2×C20 M5(2) C24 C2×C12 C20 C2×C10 C16 C2×C8 C12 C2×C6 C8 C2×C4 C4 C22 C15 C5 C3 C1 # reps 1 2 1 2 2 2 4 4 2 4 4 8 4 4 4 8 8 8 8 8 16 8 16 16 16 16 32 32 4 8 16 32

Matrix representation of C15×M5(2) in GL2(𝔽241) generated by

 54 0 0 54
,
 135 239 71 106
,
 1 0 135 240
G:=sub<GL(2,GF(241))| [54,0,0,54],[135,71,239,106],[1,135,0,240] >;

C15×M5(2) in GAP, Magma, Sage, TeX

C_{15}\times M_5(2)
% in TeX

G:=Group("C15xM5(2)");
// GroupNames label

G:=SmallGroup(480,213);
// by ID

G=gap.SmallGroup(480,213);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,420,3389,102,124]);
// 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^9>;
// generators/relations

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