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G = C51⋊D4order 408 = 23·3·17

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

C1C102 — C51⋊D4
C1C17C51C102C6×D17 — C51⋊D4
C51C102 — C51⋊D4
C1C2

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

6C2
34C2
3C22
17C22
51C4
2S3
34C6
2D17
6C34
51D4
17C2×C6
17Dic3
3Dic17
3C2×C34
2S3×C17
2C3×D17
17C3⋊D4
3C17⋊D4

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

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

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

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

57 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C17A···17H34A···34H34I···34X51A···51H102A···102H
order12223466617···1734···3434···3451···51102···102
size116342102234342···22···26···64···44···4

57 irreducible representations

dim1111222222244
type++++++++++-
imageC1C2C2C2S3D4D6C3⋊D4D17D34C17⋊D4S3×D17C51⋊D4
kernelC51⋊D4Dic51C6×D17S3×C34D34C51C34C17D6C6C3C2C1
# reps11111112881688

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

5310400
035500
0078137
00299
,
121600
23140800
00168351
00268241
,
121600
040800
00168351
00268241
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

Export

Subgroup lattice of C51⋊D4 in TeX

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