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## G = C2×C6×D17order 408 = 23·3·17

### Direct product of C2×C6 and D17

Aliases: C2×C6×D17, C513C23, C1023C22, C34⋊(C2×C6), C17⋊(C22×C6), (C2×C34)⋊5C6, (C2×C102)⋊5C2, SmallGroup(408,43)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C17 — C2×C6×D17
 Chief series C1 — C17 — C51 — C3×D17 — C6×D17 — C2×C6×D17
 Lower central C17 — C2×C6×D17
 Upper central C1 — C2×C6

Generators and relations for C2×C6×D17
G = < a,b,c,d | a2=b6=c17=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 416 in 64 conjugacy classes, 42 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C6, C23, C2×C6, C2×C6, C17, C22×C6, D17, C34, C51, D34, C2×C34, C3×D17, C102, C22×D17, C6×D17, C2×C102, C2×C6×D17
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C22×C6, D17, D34, C3×D17, C22×D17, C6×D17, C2×C6×D17

Smallest permutation representation of C2×C6×D17
On 204 points
Generators in S204
(1 170)(2 154)(3 155)(4 156)(5 157)(6 158)(7 159)(8 160)(9 161)(10 162)(11 163)(12 164)(13 165)(14 166)(15 167)(16 168)(17 169)(18 180)(19 181)(20 182)(21 183)(22 184)(23 185)(24 186)(25 187)(26 171)(27 172)(28 173)(29 174)(30 175)(31 176)(32 177)(33 178)(34 179)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 197)(42 198)(43 199)(44 200)(45 201)(46 202)(47 203)(48 204)(49 188)(50 189)(51 190)(52 119)(53 103)(54 104)(55 105)(56 106)(57 107)(58 108)(59 109)(60 110)(61 111)(62 112)(63 113)(64 114)(65 115)(66 116)(67 117)(68 118)(69 121)(70 122)(71 123)(72 124)(73 125)(74 126)(75 127)(76 128)(77 129)(78 130)(79 131)(80 132)(81 133)(82 134)(83 135)(84 136)(85 120)(86 144)(87 145)(88 146)(89 147)(90 148)(91 149)(92 150)(93 151)(94 152)(95 153)(96 137)(97 138)(98 139)(99 140)(100 141)(101 142)(102 143)
(1 69 46 66 22 102)(2 70 47 67 23 86)(3 71 48 68 24 87)(4 72 49 52 25 88)(5 73 50 53 26 89)(6 74 51 54 27 90)(7 75 35 55 28 91)(8 76 36 56 29 92)(9 77 37 57 30 93)(10 78 38 58 31 94)(11 79 39 59 32 95)(12 80 40 60 33 96)(13 81 41 61 34 97)(14 82 42 62 18 98)(15 83 43 63 19 99)(16 84 44 64 20 100)(17 85 45 65 21 101)(103 171 147 157 125 189)(104 172 148 158 126 190)(105 173 149 159 127 191)(106 174 150 160 128 192)(107 175 151 161 129 193)(108 176 152 162 130 194)(109 177 153 163 131 195)(110 178 137 164 132 196)(111 179 138 165 133 197)(112 180 139 166 134 198)(113 181 140 167 135 199)(114 182 141 168 136 200)(115 183 142 169 120 201)(116 184 143 170 121 202)(117 185 144 154 122 203)(118 186 145 155 123 204)(119 187 146 156 124 188)
(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)
(1 115)(2 114)(3 113)(4 112)(5 111)(6 110)(7 109)(8 108)(9 107)(10 106)(11 105)(12 104)(13 103)(14 119)(15 118)(16 117)(17 116)(18 124)(19 123)(20 122)(21 121)(22 120)(23 136)(24 135)(25 134)(26 133)(27 132)(28 131)(29 130)(30 129)(31 128)(32 127)(33 126)(34 125)(35 153)(36 152)(37 151)(38 150)(39 149)(40 148)(41 147)(42 146)(43 145)(44 144)(45 143)(46 142)(47 141)(48 140)(49 139)(50 138)(51 137)(52 166)(53 165)(54 164)(55 163)(56 162)(57 161)(58 160)(59 159)(60 158)(61 157)(62 156)(63 155)(64 154)(65 170)(66 169)(67 168)(68 167)(69 183)(70 182)(71 181)(72 180)(73 179)(74 178)(75 177)(76 176)(77 175)(78 174)(79 173)(80 172)(81 171)(82 187)(83 186)(84 185)(85 184)(86 200)(87 199)(88 198)(89 197)(90 196)(91 195)(92 194)(93 193)(94 192)(95 191)(96 190)(97 189)(98 188)(99 204)(100 203)(101 202)(102 201)

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

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

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

120 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A ··· 6F 6G ··· 6N 17A ··· 17H 34A ··· 34X 51A ··· 51P 102A ··· 102AV order 1 2 2 2 2 2 2 2 3 3 6 ··· 6 6 ··· 6 17 ··· 17 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 1 1 17 17 17 17 1 1 1 ··· 1 17 ··· 17 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 D17 D34 C3×D17 C6×D17 kernel C2×C6×D17 C6×D17 C2×C102 C22×D17 D34 C2×C34 C2×C6 C6 C22 C2 # reps 1 6 1 2 12 2 8 24 16 48

Matrix representation of C2×C6×D17 in GL4(𝔽103) generated by

 1 0 0 0 0 102 0 0 0 0 1 0 0 0 0 1
,
 57 0 0 0 0 102 0 0 0 0 102 0 0 0 0 102
,
 1 0 0 0 0 1 0 0 0 0 20 1 0 0 36 7
,
 1 0 0 0 0 102 0 0 0 0 81 7 0 0 34 22
G:=sub<GL(4,GF(103))| [1,0,0,0,0,102,0,0,0,0,1,0,0,0,0,1],[57,0,0,0,0,102,0,0,0,0,102,0,0,0,0,102],[1,0,0,0,0,1,0,0,0,0,20,36,0,0,1,7],[1,0,0,0,0,102,0,0,0,0,81,34,0,0,7,22] >;

C2×C6×D17 in GAP, Magma, Sage, TeX

C_2\times C_6\times D_{17}
% in TeX

G:=Group("C2xC6xD17");
// GroupNames label

G:=SmallGroup(408,43);
// by ID

G=gap.SmallGroup(408,43);
# by ID

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

G:=Group<a,b,c,d|a^2=b^6=c^17=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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