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G = C4×D51order 408 = 23·3·17

Direct product of C4 and D51

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C4×D51, C682S3, C2042C2, C122D17, C6.9D34, C34.9D6, C2.1D102, Dic515C2, D102.2C2, C102.9C22, C173(C4×S3), C517(C2×C4), C32(C4×D17), SmallGroup(408,26)

Series: Derived Chief Lower central Upper central

C1C51 — C4×D51
C1C17C51C102D102 — C4×D51
C51 — C4×D51
C1C4

Generators and relations for C4×D51
 G = < a,b,c | a4=b51=c2=1, ab=ba, ac=ca, cbc=b-1 >

51C2
51C2
51C4
51C22
17S3
17S3
3D17
3D17
51C2×C4
17Dic3
17D6
3D34
3Dic17
17C4×S3
3C4×D17

Smallest permutation representation of C4×D51
On 204 points
Generators in S204
(1 194 56 123)(2 195 57 124)(3 196 58 125)(4 197 59 126)(5 198 60 127)(6 199 61 128)(7 200 62 129)(8 201 63 130)(9 202 64 131)(10 203 65 132)(11 204 66 133)(12 154 67 134)(13 155 68 135)(14 156 69 136)(15 157 70 137)(16 158 71 138)(17 159 72 139)(18 160 73 140)(19 161 74 141)(20 162 75 142)(21 163 76 143)(22 164 77 144)(23 165 78 145)(24 166 79 146)(25 167 80 147)(26 168 81 148)(27 169 82 149)(28 170 83 150)(29 171 84 151)(30 172 85 152)(31 173 86 153)(32 174 87 103)(33 175 88 104)(34 176 89 105)(35 177 90 106)(36 178 91 107)(37 179 92 108)(38 180 93 109)(39 181 94 110)(40 182 95 111)(41 183 96 112)(42 184 97 113)(43 185 98 114)(44 186 99 115)(45 187 100 116)(46 188 101 117)(47 189 102 118)(48 190 52 119)(49 191 53 120)(50 192 54 121)(51 193 55 122)
(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 55)(2 54)(3 53)(4 52)(5 102)(6 101)(7 100)(8 99)(9 98)(10 97)(11 96)(12 95)(13 94)(14 93)(15 92)(16 91)(17 90)(18 89)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 81)(27 80)(28 79)(29 78)(30 77)(31 76)(32 75)(33 74)(34 73)(35 72)(36 71)(37 70)(38 69)(39 68)(40 67)(41 66)(42 65)(43 64)(44 63)(45 62)(46 61)(47 60)(48 59)(49 58)(50 57)(51 56)(103 162)(104 161)(105 160)(106 159)(107 158)(108 157)(109 156)(110 155)(111 154)(112 204)(113 203)(114 202)(115 201)(116 200)(117 199)(118 198)(119 197)(120 196)(121 195)(122 194)(123 193)(124 192)(125 191)(126 190)(127 189)(128 188)(129 187)(130 186)(131 185)(132 184)(133 183)(134 182)(135 181)(136 180)(137 179)(138 178)(139 177)(140 176)(141 175)(142 174)(143 173)(144 172)(145 171)(146 170)(147 169)(148 168)(149 167)(150 166)(151 165)(152 164)(153 163)

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

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

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

108 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D 6 12A12B17A···17H34A···34H51A···51P68A···68P102A···102P204A···204AF
order1222344446121217···1734···3451···5168···68102···102204···204
size11515121151512222···22···22···22···22···22···2

108 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3D6C4×S3D17D34D51C4×D17D102C4×D51
kernelC4×D51Dic51C204D102D51C68C34C17C12C6C4C3C2C1
# reps111141128816161632

Matrix representation of C4×D51 in GL2(𝔽409) generated by

2660
0266
,
19252
157145
,
390157
35219
G:=sub<GL(2,GF(409))| [266,0,0,266],[19,157,252,145],[390,352,157,19] >;

C4×D51 in GAP, Magma, Sage, TeX

C_4\times D_{51}
% in TeX

G:=Group("C4xD51");
// GroupNames label

G:=SmallGroup(408,26);
// by ID

G=gap.SmallGroup(408,26);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-17,26,323,9604]);
// Polycyclic

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

Export

Subgroup lattice of C4×D51 in TeX

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