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G = C7×D25order 350 = 2·52·7

Direct product of C7 and D25

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

Aliases: C7×D25, C25⋊C14, C1752C2, C35.2D5, C5.(C7×D5), SmallGroup(350,1)

Series: Derived Chief Lower central Upper central

C1C25 — C7×D25
C1C5C25C175 — C7×D25
C25 — C7×D25
C1C7

Generators and relations for C7×D25
 G = < a,b,c | a7=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
5D5
25C14
5C7×D5

Smallest permutation representation of C7×D25
On 175 points
Generators in S175
(1 171 142 108 76 53 46)(2 172 143 109 77 54 47)(3 173 144 110 78 55 48)(4 174 145 111 79 56 49)(5 175 146 112 80 57 50)(6 151 147 113 81 58 26)(7 152 148 114 82 59 27)(8 153 149 115 83 60 28)(9 154 150 116 84 61 29)(10 155 126 117 85 62 30)(11 156 127 118 86 63 31)(12 157 128 119 87 64 32)(13 158 129 120 88 65 33)(14 159 130 121 89 66 34)(15 160 131 122 90 67 35)(16 161 132 123 91 68 36)(17 162 133 124 92 69 37)(18 163 134 125 93 70 38)(19 164 135 101 94 71 39)(20 165 136 102 95 72 40)(21 166 137 103 96 73 41)(22 167 138 104 97 74 42)(23 168 139 105 98 75 43)(24 169 140 106 99 51 44)(25 170 141 107 100 52 45)
(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)
(1 25)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(41 50)(42 49)(43 48)(44 47)(45 46)(51 54)(52 53)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(76 100)(77 99)(78 98)(79 97)(80 96)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 125)(116 124)(117 123)(118 122)(119 121)(126 132)(127 131)(128 130)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(166 175)(167 174)(168 173)(169 172)(170 171)

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

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

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

98 conjugacy classes

class 1  2 5A5B7A···7F14A···14F25A···25J35A···35L175A···175BH
order12557···714···1425···2535···35175···175
size125221···125···252···22···22···2

98 irreducible representations

dim11112222
type++++
imageC1C2C7C14D5D25C7×D5C7×D25
kernelC7×D25C175D25C25C35C7C5C1
# reps11662101260

Matrix representation of C7×D25 in GL2(𝔽701) generated by

5500
0550
,
178472
229304
,
304676
472397
G:=sub<GL(2,GF(701))| [550,0,0,550],[178,229,472,304],[304,472,676,397] >;

C7×D25 in GAP, Magma, Sage, TeX

C_7\times D_{25}
% in TeX

G:=Group("C7xD25");
// GroupNames label

G:=SmallGroup(350,1);
// by ID

G=gap.SmallGroup(350,1);
# by ID

G:=PCGroup([4,-2,-7,-5,-5,1514,250,4483]);
// Polycyclic

G:=Group<a,b,c|a^7=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D25 in TeX

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