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G = D4×C51order 408 = 23·3·17

Direct product of C51 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C51, C4⋊C102, C683C6, C2047C2, C123C34, C222C102, C102.23C22, (C2×C34)⋊3C6, (C2×C6)⋊1C34, (C2×C102)⋊1C2, C34.6(C2×C6), C6.6(C2×C34), C2.1(C2×C102), SmallGroup(408,31)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C51
C1C2C34C102C2×C102 — D4×C51
C1C2 — D4×C51
C1C102 — D4×C51

Generators and relations for D4×C51
 G = < a,b,c | a51=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C6
2C6
2C34
2C34
2C102
2C102

Smallest permutation representation of D4×C51
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)
(1 109 73 166)(2 110 74 167)(3 111 75 168)(4 112 76 169)(5 113 77 170)(6 114 78 171)(7 115 79 172)(8 116 80 173)(9 117 81 174)(10 118 82 175)(11 119 83 176)(12 120 84 177)(13 121 85 178)(14 122 86 179)(15 123 87 180)(16 124 88 181)(17 125 89 182)(18 126 90 183)(19 127 91 184)(20 128 92 185)(21 129 93 186)(22 130 94 187)(23 131 95 188)(24 132 96 189)(25 133 97 190)(26 134 98 191)(27 135 99 192)(28 136 100 193)(29 137 101 194)(30 138 102 195)(31 139 52 196)(32 140 53 197)(33 141 54 198)(34 142 55 199)(35 143 56 200)(36 144 57 201)(37 145 58 202)(38 146 59 203)(39 147 60 204)(40 148 61 154)(41 149 62 155)(42 150 63 156)(43 151 64 157)(44 152 65 158)(45 153 66 159)(46 103 67 160)(47 104 68 161)(48 105 69 162)(49 106 70 163)(50 107 71 164)(51 108 72 165)
(103 160)(104 161)(105 162)(106 163)(107 164)(108 165)(109 166)(110 167)(111 168)(112 169)(113 170)(114 171)(115 172)(116 173)(117 174)(118 175)(119 176)(120 177)(121 178)(122 179)(123 180)(124 181)(125 182)(126 183)(127 184)(128 185)(129 186)(130 187)(131 188)(132 189)(133 190)(134 191)(135 192)(136 193)(137 194)(138 195)(139 196)(140 197)(141 198)(142 199)(143 200)(144 201)(145 202)(146 203)(147 204)(148 154)(149 155)(150 156)(151 157)(152 158)(153 159)

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

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

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

255 conjugacy classes

class 1 2A2B2C3A3B 4 6A6B6C6D6E6F12A12B17A···17P34A···34P34Q···34AV51A···51AF68A···68P102A···102AF102AG···102CR204A···204AF
order1222334666666121217···1734···3434···3451···5168···68102···102102···102204···204
size1122112112222221···11···12···21···12···21···12···22···2

255 irreducible representations

dim1111111111112222
type++++
imageC1C2C2C3C6C6C17C34C34C51C102C102D4C3×D4D4×C17D4×C51
kernelD4×C51C204C2×C102D4×C17C68C2×C34C3×D4C12C2×C6D4C4C22C51C17C3C1
# reps112224161632323264121632

Matrix representation of D4×C51 in GL2(𝔽409) generated by

3090
0309
,
542
382355
,
10
355408
G:=sub<GL(2,GF(409))| [309,0,0,309],[54,382,2,355],[1,355,0,408] >;

D4×C51 in GAP, Magma, Sage, TeX

D_4\times C_{51}
% in TeX

G:=Group("D4xC51");
// GroupNames label

G:=SmallGroup(408,31);
// by ID

G=gap.SmallGroup(408,31);
# by ID

G:=PCGroup([5,-2,-2,-3,-17,-2,2061]);
// Polycyclic

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

Export

Subgroup lattice of D4×C51 in TeX

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