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G = C10×C3⋊C8order 240 = 24·3·5

Direct product of C10 and C3⋊C8

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C10×C3⋊C8, C6⋊C40, C305C8, C12.3C20, C60.15C4, C20.58D6, C20.9Dic3, C60.75C22, C32(C2×C40), C1515(C2×C8), C6.5(C2×C20), (C2×C6).2C20, C4.14(S3×C10), (C2×C30).10C4, C30.58(C2×C4), (C2×C20).11S3, (C2×C12).6C10, (C2×C60).18C2, C4.3(C5×Dic3), C12.14(C2×C10), C2.1(C10×Dic3), (C2×C10).6Dic3, C10.18(C2×Dic3), C22.2(C5×Dic3), (C2×C4).5(C5×S3), SmallGroup(240,54)

Series: Derived Chief Lower central Upper central

C1C3 — C10×C3⋊C8
C1C3C6C12C60C5×C3⋊C8 — C10×C3⋊C8
C3 — C10×C3⋊C8
C1C2×C20

Generators and relations for C10×C3⋊C8
 G = < a,b,c | a10=b3=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

3C8
3C8
3C2×C8
3C40
3C40
3C2×C40

Smallest permutation representation of C10×C3⋊C8
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)
(1 221 121)(2 222 122)(3 223 123)(4 224 124)(5 225 125)(6 226 126)(7 227 127)(8 228 128)(9 229 129)(10 230 130)(11 105 31)(12 106 32)(13 107 33)(14 108 34)(15 109 35)(16 110 36)(17 101 37)(18 102 38)(19 103 39)(20 104 40)(21 89 75)(22 90 76)(23 81 77)(24 82 78)(25 83 79)(26 84 80)(27 85 71)(28 86 72)(29 87 73)(30 88 74)(41 240 65)(42 231 66)(43 232 67)(44 233 68)(45 234 69)(46 235 70)(47 236 61)(48 237 62)(49 238 63)(50 239 64)(51 181 135)(52 182 136)(53 183 137)(54 184 138)(55 185 139)(56 186 140)(57 187 131)(58 188 132)(59 189 133)(60 190 134)(91 146 197)(92 147 198)(93 148 199)(94 149 200)(95 150 191)(96 141 192)(97 142 193)(98 143 194)(99 144 195)(100 145 196)(111 207 151)(112 208 152)(113 209 153)(114 210 154)(115 201 155)(116 202 156)(117 203 157)(118 204 158)(119 205 159)(120 206 160)(161 215 171)(162 216 172)(163 217 173)(164 218 174)(165 219 175)(166 220 176)(167 211 177)(168 212 178)(169 213 179)(170 214 180)
(1 25 163 67 157 144 185 105)(2 26 164 68 158 145 186 106)(3 27 165 69 159 146 187 107)(4 28 166 70 160 147 188 108)(5 29 167 61 151 148 189 109)(6 30 168 62 152 149 190 110)(7 21 169 63 153 150 181 101)(8 22 170 64 154 141 182 102)(9 23 161 65 155 142 183 103)(10 24 162 66 156 143 184 104)(11 221 79 217 232 117 99 139)(12 222 80 218 233 118 100 140)(13 223 71 219 234 119 91 131)(14 224 72 220 235 120 92 132)(15 225 73 211 236 111 93 133)(16 226 74 212 237 112 94 134)(17 227 75 213 238 113 95 135)(18 228 76 214 239 114 96 136)(19 229 77 215 240 115 97 137)(20 230 78 216 231 116 98 138)(31 121 83 173 43 203 195 55)(32 122 84 174 44 204 196 56)(33 123 85 175 45 205 197 57)(34 124 86 176 46 206 198 58)(35 125 87 177 47 207 199 59)(36 126 88 178 48 208 200 60)(37 127 89 179 49 209 191 51)(38 128 90 180 50 210 192 52)(39 129 81 171 41 201 193 53)(40 130 82 172 42 202 194 54)

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

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

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

C10×C3⋊C8 is a maximal subgroup of
Dic154C8  C30.21C42  C30.23C42  C60.93D4  D304C8  C6.D40  D6012C4  C6.Dic20  Dic3012C4  C60.13Q8  C60.14Q8  C60.Q8  C60.5Q8  C12.59D20  Dic3×C40  D20.3Dic3  D60.4C4  D20.31D6  S3×C2×C40

120 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B5C5D6A6B6C8A···8H10A···10L12A12B12C12D15A15B15C15D20A···20P30A···30L40A···40AF60A···60P
order12223444455556668···810···10121212121515151520···2030···3040···4060···60
size11112111111112223···31···1222222221···12···23···32···2

120 irreducible representations

dim1111111111112222222222
type++++-+-
imageC1C2C2C4C4C5C8C10C10C20C20C40S3Dic3D6Dic3C3⋊C8C5×S3C5×Dic3S3×C10C5×Dic3C5×C3⋊C8
kernelC10×C3⋊C8C5×C3⋊C8C2×C60C60C2×C30C2×C3⋊C8C30C3⋊C8C2×C12C12C2×C6C6C2×C20C20C20C2×C10C10C2×C4C4C4C22C2
# reps121224884883211114444416

Matrix representation of C10×C3⋊C8 in GL3(𝔽241) generated by

24000
01430
00143
,
100
00240
01240
,
100
02338
008
G:=sub<GL(3,GF(241))| [240,0,0,0,143,0,0,0,143],[1,0,0,0,0,1,0,240,240],[1,0,0,0,233,0,0,8,8] >;

C10×C3⋊C8 in GAP, Magma, Sage, TeX

C_{10}\times C_3\rtimes C_8
% in TeX

G:=Group("C10xC3:C8");
// GroupNames label

G:=SmallGroup(240,54);
// by ID

G=gap.SmallGroup(240,54);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,120,69,5765]);
// Polycyclic

G:=Group<a,b,c|a^10=b^3=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C10×C3⋊C8 in TeX

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