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## G = C51⋊D4order 408 = 23·3·17

### 1st semidirect product of C51 and D4 acting via D4/C2=C22

Aliases: C511D4, D61D17, D341S3, C6.4D34, C34.4D6, Dic514C2, C102.4C22, (S3×C34)⋊1C2, (C6×D17)⋊1C2, C172(C3⋊D4), C32(C17⋊D4), C2.4(S3×D17), SmallGroup(408,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C102 — C51⋊D4
 Chief series C1 — C17 — C51 — C102 — C6×D17 — C51⋊D4
 Lower central C51 — C102 — C51⋊D4
 Upper central C1 — C2

Generators and relations for C51⋊D4
G = < a,b,c | a51=b4=c2=1, bab-1=a-1, cac=a16, cbc=b-1 >

Smallest permutation representation of C51⋊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)
(1 135 80 166)(2 134 81 165)(3 133 82 164)(4 132 83 163)(5 131 84 162)(6 130 85 161)(7 129 86 160)(8 128 87 159)(9 127 88 158)(10 126 89 157)(11 125 90 156)(12 124 91 155)(13 123 92 154)(14 122 93 204)(15 121 94 203)(16 120 95 202)(17 119 96 201)(18 118 97 200)(19 117 98 199)(20 116 99 198)(21 115 100 197)(22 114 101 196)(23 113 102 195)(24 112 52 194)(25 111 53 193)(26 110 54 192)(27 109 55 191)(28 108 56 190)(29 107 57 189)(30 106 58 188)(31 105 59 187)(32 104 60 186)(33 103 61 185)(34 153 62 184)(35 152 63 183)(36 151 64 182)(37 150 65 181)(38 149 66 180)(39 148 67 179)(40 147 68 178)(41 146 69 177)(42 145 70 176)(43 144 71 175)(44 143 72 174)(45 142 73 173)(46 141 74 172)(47 140 75 171)(48 139 76 170)(49 138 77 169)(50 137 78 168)(51 136 79 167)
(2 17)(3 33)(4 49)(5 14)(6 30)(7 46)(8 11)(9 27)(10 43)(12 24)(13 40)(15 21)(16 37)(19 34)(20 50)(22 31)(23 47)(25 28)(26 44)(29 41)(32 38)(36 51)(39 48)(42 45)(52 91)(53 56)(54 72)(55 88)(57 69)(58 85)(59 101)(60 66)(61 82)(62 98)(64 79)(65 95)(67 76)(68 92)(70 73)(71 89)(74 86)(75 102)(77 83)(78 99)(81 96)(84 93)(87 90)(94 100)(103 164)(104 180)(105 196)(106 161)(107 177)(108 193)(109 158)(110 174)(111 190)(112 155)(113 171)(114 187)(115 203)(116 168)(117 184)(118 200)(119 165)(120 181)(121 197)(122 162)(123 178)(124 194)(125 159)(126 175)(127 191)(128 156)(129 172)(130 188)(131 204)(132 169)(133 185)(134 201)(135 166)(136 182)(137 198)(138 163)(139 179)(140 195)(141 160)(142 176)(143 192)(144 157)(145 173)(146 189)(147 154)(148 170)(149 186)(150 202)(151 167)(152 183)(153 199)

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

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

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

57 conjugacy classes

 class 1 2A 2B 2C 3 4 6A 6B 6C 17A ··· 17H 34A ··· 34H 34I ··· 34X 51A ··· 51H 102A ··· 102H order 1 2 2 2 3 4 6 6 6 17 ··· 17 34 ··· 34 34 ··· 34 51 ··· 51 102 ··· 102 size 1 1 6 34 2 102 2 34 34 2 ··· 2 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

57 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 S3 D4 D6 C3⋊D4 D17 D34 C17⋊D4 S3×D17 C51⋊D4 kernel C51⋊D4 Dic51 C6×D17 S3×C34 D34 C51 C34 C17 D6 C6 C3 C2 C1 # reps 1 1 1 1 1 1 1 2 8 8 16 8 8

Matrix representation of C51⋊D4 in GL4(𝔽409) generated by

 53 104 0 0 0 355 0 0 0 0 78 137 0 0 29 9
,
 1 216 0 0 231 408 0 0 0 0 168 351 0 0 268 241
,
 1 216 0 0 0 408 0 0 0 0 168 351 0 0 268 241
G:=sub<GL(4,GF(409))| [53,0,0,0,104,355,0,0,0,0,78,29,0,0,137,9],[1,231,0,0,216,408,0,0,0,0,168,268,0,0,351,241],[1,0,0,0,216,408,0,0,0,0,168,268,0,0,351,241] >;

C51⋊D4 in GAP, Magma, Sage, TeX

C_{51}\rtimes D_4
% in TeX

G:=Group("C51:D4");
// GroupNames label

G:=SmallGroup(408,10);
// by ID

G=gap.SmallGroup(408,10);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,61,168,9604]);
// Polycyclic

G:=Group<a,b,c|a^51=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^16,c*b*c=b^-1>;
// generators/relations

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