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## G = C15×Q16order 240 = 24·3·5

### Direct product of C15 and Q16

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

Aliases: C15×Q16, C8.C30, C40.3C6, C120.6C2, C24.2C10, C30.56D4, Q8.2C30, C60.79C22, C4.3(C2×C30), C2.5(D4×C15), C6.16(C5×D4), (C5×Q8).4C6, C20.19(C2×C6), C10.16(C3×D4), (C3×Q8).2C10, (Q8×C15).4C2, C12.19(C2×C10), SmallGroup(240,88)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C15×Q16
 Chief series C1 — C2 — C4 — C20 — C60 — Q8×C15 — C15×Q16
 Lower central C1 — C2 — C4 — C15×Q16
 Upper central C1 — C30 — C60 — C15×Q16

Generators and relations for C15×Q16
G = < a,b,c | a15=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C15×Q16
Regular action 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 28 218 184 122 90 174 149)(2 29 219 185 123 76 175 150)(3 30 220 186 124 77 176 136)(4 16 221 187 125 78 177 137)(5 17 222 188 126 79 178 138)(6 18 223 189 127 80 179 139)(7 19 224 190 128 81 180 140)(8 20 225 191 129 82 166 141)(9 21 211 192 130 83 167 142)(10 22 212 193 131 84 168 143)(11 23 213 194 132 85 169 144)(12 24 214 195 133 86 170 145)(13 25 215 181 134 87 171 146)(14 26 216 182 135 88 172 147)(15 27 217 183 121 89 173 148)(31 47 210 71 117 155 101 235)(32 48 196 72 118 156 102 236)(33 49 197 73 119 157 103 237)(34 50 198 74 120 158 104 238)(35 51 199 75 106 159 105 239)(36 52 200 61 107 160 91 240)(37 53 201 62 108 161 92 226)(38 54 202 63 109 162 93 227)(39 55 203 64 110 163 94 228)(40 56 204 65 111 164 95 229)(41 57 205 66 112 165 96 230)(42 58 206 67 113 151 97 231)(43 59 207 68 114 152 98 232)(44 60 208 69 115 153 99 233)(45 46 209 70 116 154 100 234)
(1 70 122 234)(2 71 123 235)(3 72 124 236)(4 73 125 237)(5 74 126 238)(6 75 127 239)(7 61 128 240)(8 62 129 226)(9 63 130 227)(10 64 131 228)(11 65 132 229)(12 66 133 230)(13 67 134 231)(14 68 135 232)(15 69 121 233)(16 197 78 103)(17 198 79 104)(18 199 80 105)(19 200 81 91)(20 201 82 92)(21 202 83 93)(22 203 84 94)(23 204 85 95)(24 205 86 96)(25 206 87 97)(26 207 88 98)(27 208 89 99)(28 209 90 100)(29 210 76 101)(30 196 77 102)(31 150 117 185)(32 136 118 186)(33 137 119 187)(34 138 120 188)(35 139 106 189)(36 140 107 190)(37 141 108 191)(38 142 109 192)(39 143 110 193)(40 144 111 194)(41 145 112 195)(42 146 113 181)(43 147 114 182)(44 148 115 183)(45 149 116 184)(46 174 154 218)(47 175 155 219)(48 176 156 220)(49 177 157 221)(50 178 158 222)(51 179 159 223)(52 180 160 224)(53 166 161 225)(54 167 162 211)(55 168 163 212)(56 169 164 213)(57 170 165 214)(58 171 151 215)(59 172 152 216)(60 173 153 217)

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

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

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

C15×Q16 is a maximal subgroup of   C8.6D30  C157Q32  Q16⋊D15  D1208C2

105 conjugacy classes

 class 1 2 3A 3B 4A 4B 4C 5A 5B 5C 5D 6A 6B 8A 8B 10A 10B 10C 10D 12A 12B 12C 12D 12E 12F 15A ··· 15H 20A 20B 20C 20D 20E ··· 20L 24A 24B 24C 24D 30A ··· 30H 40A ··· 40H 60A ··· 60H 60I ··· 60X 120A ··· 120P order 1 2 3 3 4 4 4 5 5 5 5 6 6 8 8 10 10 10 10 12 12 12 12 12 12 15 ··· 15 20 20 20 20 20 ··· 20 24 24 24 24 30 ··· 30 40 ··· 40 60 ··· 60 60 ··· 60 120 ··· 120 size 1 1 1 1 2 4 4 1 1 1 1 1 1 2 2 1 1 1 1 2 2 4 4 4 4 1 ··· 1 2 2 2 2 4 ··· 4 2 2 2 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 2 ··· 2

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 D4 Q16 C3×D4 C5×D4 C3×Q16 C5×Q16 D4×C15 C15×Q16 kernel C15×Q16 C120 Q8×C15 C5×Q16 C3×Q16 C40 C5×Q8 C24 C3×Q8 Q16 C8 Q8 C30 C15 C10 C6 C5 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 2 2 4 4 8 8 16

Matrix representation of C15×Q16 in GL2(𝔽31) generated by

 10 0 0 10
,
 7 7 7 16
,
 0 30 1 0
G:=sub<GL(2,GF(31))| [10,0,0,10],[7,7,7,16],[0,1,30,0] >;

C15×Q16 in GAP, Magma, Sage, TeX

C_{15}\times Q_{16}
% in TeX

G:=Group("C15xQ16");
// GroupNames label

G:=SmallGroup(240,88);
// by ID

G=gap.SmallGroup(240,88);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-2,720,745,727,5404,2710,88]);
// Polycyclic

G:=Group<a,b,c|a^15=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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