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G = C17×C3⋊D4order 408 = 23·3·17

Direct product of C17 and C3⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C17×C3⋊D4, C519D4, D62C34, Dic3⋊C34, C34.17D6, C102.22C22, C32(D4×C17), (C2×C6)⋊2C34, (C2×C34)⋊3S3, (S3×C34)⋊5C2, (C2×C102)⋊6C2, C2.5(S3×C34), C6.5(C2×C34), C222(S3×C17), (Dic3×C17)⋊4C2, SmallGroup(408,24)

Series: Derived Chief Lower central Upper central

C1C6 — C17×C3⋊D4
C1C3C6C102S3×C34 — C17×C3⋊D4
C3C6 — C17×C3⋊D4
C1C34C2×C34

Generators and relations for C17×C3⋊D4
 G = < a,b,c,d | a17=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

2C2
6C2
3C22
3C4
2C6
2S3
2C34
6C34
3D4
3C2×C34
3C68
2C102
2S3×C17
3D4×C17

Smallest permutation representation of C17×C3⋊D4
On 204 points
Generators in S204
(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 45 19)(2 46 20)(3 47 21)(4 48 22)(5 49 23)(6 50 24)(7 51 25)(8 35 26)(9 36 27)(10 37 28)(11 38 29)(12 39 30)(13 40 31)(14 41 32)(15 42 33)(16 43 34)(17 44 18)(52 168 98)(53 169 99)(54 170 100)(55 154 101)(56 155 102)(57 156 86)(58 157 87)(59 158 88)(60 159 89)(61 160 90)(62 161 91)(63 162 92)(64 163 93)(65 164 94)(66 165 95)(67 166 96)(68 167 97)(69 145 200)(70 146 201)(71 147 202)(72 148 203)(73 149 204)(74 150 188)(75 151 189)(76 152 190)(77 153 191)(78 137 192)(79 138 193)(80 139 194)(81 140 195)(82 141 196)(83 142 197)(84 143 198)(85 144 199)(103 134 184)(104 135 185)(105 136 186)(106 120 187)(107 121 171)(108 122 172)(109 123 173)(110 124 174)(111 125 175)(112 126 176)(113 127 177)(114 128 178)(115 129 179)(116 130 180)(117 131 181)(118 132 182)(119 133 183)
(1 188 114 100)(2 189 115 101)(3 190 116 102)(4 191 117 86)(5 192 118 87)(6 193 119 88)(7 194 103 89)(8 195 104 90)(9 196 105 91)(10 197 106 92)(11 198 107 93)(12 199 108 94)(13 200 109 95)(14 201 110 96)(15 202 111 97)(16 203 112 98)(17 204 113 99)(18 73 177 53)(19 74 178 54)(20 75 179 55)(21 76 180 56)(22 77 181 57)(23 78 182 58)(24 79 183 59)(25 80 184 60)(26 81 185 61)(27 82 186 62)(28 83 187 63)(29 84 171 64)(30 85 172 65)(31 69 173 66)(32 70 174 67)(33 71 175 68)(34 72 176 52)(35 140 135 160)(36 141 136 161)(37 142 120 162)(38 143 121 163)(39 144 122 164)(40 145 123 165)(41 146 124 166)(42 147 125 167)(43 148 126 168)(44 149 127 169)(45 150 128 170)(46 151 129 154)(47 152 130 155)(48 153 131 156)(49 137 132 157)(50 138 133 158)(51 139 134 159)
(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 35)(27 36)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(52 148)(53 149)(54 150)(55 151)(56 152)(57 153)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 165)(70 166)(71 167)(72 168)(73 169)(74 170)(75 154)(76 155)(77 156)(78 157)(79 158)(80 159)(81 160)(82 161)(83 162)(84 163)(85 164)(86 191)(87 192)(88 193)(89 194)(90 195)(91 196)(92 197)(93 198)(94 199)(95 200)(96 201)(97 202)(98 203)(99 204)(100 188)(101 189)(102 190)(120 187)(121 171)(122 172)(123 173)(124 174)(125 175)(126 176)(127 177)(128 178)(129 179)(130 180)(131 181)(132 182)(133 183)(134 184)(135 185)(136 186)

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

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

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

153 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C17A···17P34A···34P34Q···34AF34AG···34AV51A···51P68A···68P102A···102AV
order12223466617···1734···3434···3434···3451···5168···68102···102
size1126262221···11···12···26···62···26···62···2

153 irreducible representations

dim1111111122222222
type+++++++
imageC1C2C2C2C17C34C34C34S3D4D6C3⋊D4S3×C17D4×C17S3×C34C17×C3⋊D4
kernelC17×C3⋊D4Dic3×C17S3×C34C2×C102C3⋊D4Dic3D6C2×C6C2×C34C51C34C17C22C3C2C1
# reps111116161616111216161632

Matrix representation of C17×C3⋊D4 in GL2(𝔽409) generated by

1500
0150
,
408408
10
,
23765
237172
,
10
408408
G:=sub<GL(2,GF(409))| [150,0,0,150],[408,1,408,0],[237,237,65,172],[1,408,0,408] >;

C17×C3⋊D4 in GAP, Magma, Sage, TeX

C_{17}\times C_3\rtimes D_4
% in TeX

G:=Group("C17xC3:D4");
// GroupNames label

G:=SmallGroup(408,24);
// by ID

G=gap.SmallGroup(408,24);
# by ID

G:=PCGroup([5,-2,-2,-17,-2,-3,701,6804]);
// Polycyclic

G:=Group<a,b,c,d|a^17=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C17×C3⋊D4 in TeX

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