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G = C12×D17order 408 = 23·3·17

Direct product of C12 and D17

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

Aliases: C12×D17, C682C6, C2044C2, D34.2C6, C6.14D34, Dic172C6, C102.14C22, C518(C2×C4), C172(C2×C12), C34.2(C2×C6), C2.1(C6×D17), (C6×D17).4C2, (C3×Dic17)⋊5C2, SmallGroup(408,16)

Series: Derived Chief Lower central Upper central

C1C17 — C12×D17
C1C17C34C102C6×D17 — C12×D17
C17 — C12×D17
C1C12

Generators and relations for C12×D17
 G = < a,b,c | a12=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C2
17C4
17C22
17C6
17C6
17C2×C4
17C12
17C2×C6
17C2×C12

Smallest permutation representation of C12×D17
On 204 points
Generators in S204
(1 197 65 107 83 164 27 183 35 124 87 137)(2 198 66 108 84 165 28 184 36 125 88 138)(3 199 67 109 85 166 29 185 37 126 89 139)(4 200 68 110 69 167 30 186 38 127 90 140)(5 201 52 111 70 168 31 187 39 128 91 141)(6 202 53 112 71 169 32 171 40 129 92 142)(7 203 54 113 72 170 33 172 41 130 93 143)(8 204 55 114 73 154 34 173 42 131 94 144)(9 188 56 115 74 155 18 174 43 132 95 145)(10 189 57 116 75 156 19 175 44 133 96 146)(11 190 58 117 76 157 20 176 45 134 97 147)(12 191 59 118 77 158 21 177 46 135 98 148)(13 192 60 119 78 159 22 178 47 136 99 149)(14 193 61 103 79 160 23 179 48 120 100 150)(15 194 62 104 80 161 24 180 49 121 101 151)(16 195 63 105 81 162 25 181 50 122 102 152)(17 196 64 106 82 163 26 182 51 123 86 153)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(52 60)(53 59)(54 58)(55 57)(61 68)(62 67)(63 66)(64 65)(69 79)(70 78)(71 77)(72 76)(73 75)(80 85)(81 84)(82 83)(86 87)(88 102)(89 101)(90 100)(91 99)(92 98)(93 97)(94 96)(103 110)(104 109)(105 108)(106 107)(111 119)(112 118)(113 117)(114 116)(120 127)(121 126)(122 125)(123 124)(128 136)(129 135)(130 134)(131 133)(137 153)(138 152)(139 151)(140 150)(141 149)(142 148)(143 147)(144 146)(154 156)(157 170)(158 169)(159 168)(160 167)(161 166)(162 165)(163 164)(171 177)(172 176)(173 175)(178 187)(179 186)(180 185)(181 184)(182 183)(189 204)(190 203)(191 202)(192 201)(193 200)(194 199)(195 198)(196 197)

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

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

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

120 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A6B6C6D6E6F12A12B12C12D12E12F12G12H17A···17H34A···34H51A···51P68A···68P102A···102P204A···204AF
order1222334444666666121212121212121217···1734···3451···5168···68102···102204···204
size1117171111171711171717171111171717172···22···22···22···22···22···2

120 irreducible representations

dim1111111111222222
type++++++
imageC1C2C2C2C3C4C6C6C6C12D17D34C3×D17C4×D17C6×D17C12×D17
kernelC12×D17C3×Dic17C204C6×D17C4×D17C3×D17Dic17C68D34D17C12C6C4C3C2C1
# reps11112422288816161632

Matrix representation of C12×D17 in GL3(𝔽409) generated by

14300
03560
00356
,
100
0304119
0408266
,
40800
0134382
0256275
G:=sub<GL(3,GF(409))| [143,0,0,0,356,0,0,0,356],[1,0,0,0,304,408,0,119,266],[408,0,0,0,134,256,0,382,275] >;

C12×D17 in GAP, Magma, Sage, TeX

C_{12}\times D_{17}
% in TeX

G:=Group("C12xD17");
// GroupNames label

G:=SmallGroup(408,16);
// by ID

G=gap.SmallGroup(408,16);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-17,66,9604]);
// Polycyclic

G:=Group<a,b,c|a^12=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C12×D17 in TeX

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