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G = C15×C8.C4order 480 = 25·3·5

Direct product of C15 and C8.C4

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

Aliases: C15×C8.C4, C8.1C60, C24.2C20, C120.23C4, C40.10C12, C60.253D4, M4(2).2C30, C4.8(C2×C60), (C2×C8).5C30, (C2×C40).15C6, C22.(Q8×C15), C20.68(C3×D4), C12.68(C5×D4), C4.19(D4×C15), C30.66(C4⋊C4), (C2×C30).11Q8, C60.259(C2×C4), (C2×C120).31C2, C20.66(C2×C12), C12.45(C2×C20), (C2×C24).11C10, (C5×M4(2)).4C6, (C2×C60).574C22, (C3×M4(2)).4C10, (C15×M4(2)).8C2, C2.5(C15×C4⋊C4), C6.14(C5×C4⋊C4), C10.21(C3×C4⋊C4), (C2×C6).2(C5×Q8), (C2×C10).2(C3×Q8), (C2×C4).21(C2×C30), (C2×C20).122(C2×C6), (C2×C12).125(C2×C10), SmallGroup(480,211)

Series: Derived Chief Lower central Upper central

C1C4 — C15×C8.C4
C1C2C4C2×C4C2×C20C2×C60C15×M4(2) — C15×C8.C4
C1C2C4 — C15×C8.C4
C1C60C2×C60 — C15×C8.C4

Generators and relations for C15×C8.C4
 G = < a,b,c | a15=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 72 in 60 conjugacy classes, 48 normal (40 characteristic)
C1, C2, C2, C3, C4 [×2], C22, C5, C6, C6, C8 [×2], C8 [×2], C2×C4, C10, C10, C12 [×2], C2×C6, C15, C2×C8, M4(2) [×2], C20 [×2], C2×C10, C24 [×2], C24 [×2], C2×C12, C30, C30, C8.C4, C40 [×2], C40 [×2], C2×C20, C2×C24, C3×M4(2) [×2], C60 [×2], C2×C30, C2×C40, C5×M4(2) [×2], C3×C8.C4, C120 [×2], C120 [×2], C2×C60, C5×C8.C4, C2×C120, C15×M4(2) [×2], C15×C8.C4
Quotients: C1, C2 [×3], C3, C4 [×2], C22, C5, C6 [×3], C2×C4, D4, Q8, C10 [×3], C12 [×2], C2×C6, C15, C4⋊C4, C20 [×2], C2×C10, C2×C12, C3×D4, C3×Q8, C30 [×3], C8.C4, C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C60 [×2], C2×C30, C5×C4⋊C4, C3×C8.C4, C2×C60, D4×C15, Q8×C15, C5×C8.C4, C15×C4⋊C4, C15×C8.C4

Smallest permutation representation of C15×C8.C4
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 89 218 115 228 153 94 122)(2 90 219 116 229 154 95 123)(3 76 220 117 230 155 96 124)(4 77 221 118 231 156 97 125)(5 78 222 119 232 157 98 126)(6 79 223 120 233 158 99 127)(7 80 224 106 234 159 100 128)(8 81 225 107 235 160 101 129)(9 82 211 108 236 161 102 130)(10 83 212 109 237 162 103 131)(11 84 213 110 238 163 104 132)(12 85 214 111 239 164 105 133)(13 86 215 112 240 165 91 134)(14 87 216 113 226 151 92 135)(15 88 217 114 227 152 93 121)(16 53 42 146 61 171 208 190)(17 54 43 147 62 172 209 191)(18 55 44 148 63 173 210 192)(19 56 45 149 64 174 196 193)(20 57 31 150 65 175 197 194)(21 58 32 136 66 176 198 195)(22 59 33 137 67 177 199 181)(23 60 34 138 68 178 200 182)(24 46 35 139 69 179 201 183)(25 47 36 140 70 180 202 184)(26 48 37 141 71 166 203 185)(27 49 38 142 72 167 204 186)(28 50 39 143 73 168 205 187)(29 51 40 144 74 169 206 188)(30 52 41 145 75 170 207 189)
(1 42 218 16 228 208 94 61)(2 43 219 17 229 209 95 62)(3 44 220 18 230 210 96 63)(4 45 221 19 231 196 97 64)(5 31 222 20 232 197 98 65)(6 32 223 21 233 198 99 66)(7 33 224 22 234 199 100 67)(8 34 225 23 235 200 101 68)(9 35 211 24 236 201 102 69)(10 36 212 25 237 202 103 70)(11 37 213 26 238 203 104 71)(12 38 214 27 239 204 105 72)(13 39 215 28 240 205 91 73)(14 40 216 29 226 206 92 74)(15 41 217 30 227 207 93 75)(46 108 183 161 179 130 139 82)(47 109 184 162 180 131 140 83)(48 110 185 163 166 132 141 84)(49 111 186 164 167 133 142 85)(50 112 187 165 168 134 143 86)(51 113 188 151 169 135 144 87)(52 114 189 152 170 121 145 88)(53 115 190 153 171 122 146 89)(54 116 191 154 172 123 147 90)(55 117 192 155 173 124 148 76)(56 118 193 156 174 125 149 77)(57 119 194 157 175 126 150 78)(58 120 195 158 176 127 136 79)(59 106 181 159 177 128 137 80)(60 107 182 160 178 129 138 81)

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

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

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

210 conjugacy classes

class 1 2A2B3A3B4A4B4C5A5B5C5D6A6B6C6D8A8B8C8D8E8F8G8H10A10B10C10D10E10F10G10H12A12B12C12D12E12F15A···15H20A···20H20I20J20K20L24A···24H24I···24P30A···30H30I···30P40A···40P40Q···40AF60A···60P60Q···60X120A···120AF120AG···120BL
order122334445555666688888888101010101010101012121212121215···1520···202020202024···2424···2430···3030···3040···4040···4060···6060···60120···120120···120
size112111121111112222224444111122221111221···11···122222···24···41···12···22···24···41···12···22···24···4

210 irreducible representations

dim1111111111111111222222222222
type++++-
imageC1C2C2C3C4C5C6C6C10C10C12C15C20C30C30C60D4Q8C3×D4C3×Q8C8.C4C5×D4C5×Q8C3×C8.C4D4×C15Q8×C15C5×C8.C4C15×C8.C4
kernelC15×C8.C4C2×C120C15×M4(2)C5×C8.C4C120C3×C8.C4C2×C40C5×M4(2)C2×C24C3×M4(2)C40C8.C4C24C2×C8M4(2)C8C60C2×C30C20C2×C10C15C12C2×C6C5C4C22C3C1
# reps112244244888168163211224448881632

Matrix representation of C15×C8.C4 in GL3(𝔽241) generated by

1500
0910
0091
,
24000
0211129
008
,
17700
054165
0128187
G:=sub<GL(3,GF(241))| [15,0,0,0,91,0,0,0,91],[240,0,0,0,211,0,0,129,8],[177,0,0,0,54,128,0,165,187] >;

C15×C8.C4 in GAP, Magma, Sage, TeX

C_{15}\times C_8.C_4
% in TeX

G:=Group("C15xC8.C4");
// GroupNames label

G:=SmallGroup(480,211);
// by ID

G=gap.SmallGroup(480,211);
# by ID

G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,428,10504,172,124]);
// Polycyclic

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

׿
×
𝔽