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G = C517D4order 408 = 23·3·17

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C517D4, D1022C2, C6.12D34, C34.12D6, C2.5D102, C222D51, Dic511C2, C102.12C22, (C2×C34)⋊4S3, (C2×C6)⋊2D17, (C2×C102)⋊2C2, C33(C17⋊D4), C173(C3⋊D4), SmallGroup(408,29)

Series: Derived Chief Lower central Upper central

C1C102 — C517D4
C1C17C51C102D102 — C517D4
C51C102 — C517D4
C1C2C22

Generators and relations for C517D4
 G = < a,b,c | a51=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
102C2
51C22
51C4
2C6
34S3
2C34
6D17
51D4
17D6
17Dic3
3D34
3Dic17
2C102
2D51
17C3⋊D4
3C17⋊D4

Smallest permutation representation of C517D4
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)
(1 115 88 178)(2 114 89 177)(3 113 90 176)(4 112 91 175)(5 111 92 174)(6 110 93 173)(7 109 94 172)(8 108 95 171)(9 107 96 170)(10 106 97 169)(11 105 98 168)(12 104 99 167)(13 103 100 166)(14 153 101 165)(15 152 102 164)(16 151 52 163)(17 150 53 162)(18 149 54 161)(19 148 55 160)(20 147 56 159)(21 146 57 158)(22 145 58 157)(23 144 59 156)(24 143 60 155)(25 142 61 154)(26 141 62 204)(27 140 63 203)(28 139 64 202)(29 138 65 201)(30 137 66 200)(31 136 67 199)(32 135 68 198)(33 134 69 197)(34 133 70 196)(35 132 71 195)(36 131 72 194)(37 130 73 193)(38 129 74 192)(39 128 75 191)(40 127 76 190)(41 126 77 189)(42 125 78 188)(43 124 79 187)(44 123 80 186)(45 122 81 185)(46 121 82 184)(47 120 83 183)(48 119 84 182)(49 118 85 181)(50 117 86 180)(51 116 87 179)
(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)(52 73)(53 72)(54 71)(55 70)(56 69)(57 68)(58 67)(59 66)(60 65)(61 64)(62 63)(74 102)(75 101)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(103 190)(104 189)(105 188)(106 187)(107 186)(108 185)(109 184)(110 183)(111 182)(112 181)(113 180)(114 179)(115 178)(116 177)(117 176)(118 175)(119 174)(120 173)(121 172)(122 171)(123 170)(124 169)(125 168)(126 167)(127 166)(128 165)(129 164)(130 163)(131 162)(132 161)(133 160)(134 159)(135 158)(136 157)(137 156)(138 155)(139 154)(140 204)(141 203)(142 202)(143 201)(144 200)(145 199)(146 198)(147 197)(148 196)(149 195)(150 194)(151 193)(152 192)(153 191)

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

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

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

105 conjugacy classes

class 1 2A2B2C 3  4 6A6B6C17A···17H34A···34X51A···51P102A···102AV
order12223466617···1734···3451···51102···102
size11210221022222···22···22···22···2

105 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2S3D4D6C3⋊D4D17D34D51C17⋊D4D102C517D4
kernelC517D4Dic51D102C2×C102C2×C34C51C34C17C2×C6C6C22C3C2C1
# reps111111128816161632

Matrix representation of C517D4 in GL2(𝔽409) generated by

221157
224198
,
356216
21853
,
14694
204263
G:=sub<GL(2,GF(409))| [221,224,157,198],[356,218,216,53],[146,204,94,263] >;

C517D4 in GAP, Magma, Sage, TeX

C_{51}\rtimes_7D_4
% in TeX

G:=Group("C51:7D4");
// GroupNames label

G:=SmallGroup(408,29);
// by ID

G=gap.SmallGroup(408,29);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,61,323,9604]);
// Polycyclic

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

Export

Subgroup lattice of C517D4 in TeX

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