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G = D7×C25order 350 = 2·52·7

Direct product of C25 and D7

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

Aliases: D7×C25, C7⋊C50, C1753C2, C35.C10, C5.(C5×D7), (C5×D7).C5, SmallGroup(350,2)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C25
C1C7C35C175 — D7×C25
C7 — D7×C25
C1C25

Generators and relations for D7×C25
 G = < a,b,c | a25=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C10
7C50

Smallest permutation representation of D7×C25
On 175 points
Generators in S175
(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 42 74 110 128 153 93)(2 43 75 111 129 154 94)(3 44 51 112 130 155 95)(4 45 52 113 131 156 96)(5 46 53 114 132 157 97)(6 47 54 115 133 158 98)(7 48 55 116 134 159 99)(8 49 56 117 135 160 100)(9 50 57 118 136 161 76)(10 26 58 119 137 162 77)(11 27 59 120 138 163 78)(12 28 60 121 139 164 79)(13 29 61 122 140 165 80)(14 30 62 123 141 166 81)(15 31 63 124 142 167 82)(16 32 64 125 143 168 83)(17 33 65 101 144 169 84)(18 34 66 102 145 170 85)(19 35 67 103 146 171 86)(20 36 68 104 147 172 87)(21 37 69 105 148 173 88)(22 38 70 106 149 174 89)(23 39 71 107 150 175 90)(24 40 72 108 126 151 91)(25 41 73 109 127 152 92)
(1 93)(2 94)(3 95)(4 96)(5 97)(6 98)(7 99)(8 100)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 162)(27 163)(28 164)(29 165)(30 166)(31 167)(32 168)(33 169)(34 170)(35 171)(36 172)(37 173)(38 174)(39 175)(40 151)(41 152)(42 153)(43 154)(44 155)(45 156)(46 157)(47 158)(48 159)(49 160)(50 161)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 139)(61 140)(62 141)(63 142)(64 143)(65 144)(66 145)(67 146)(68 147)(69 148)(70 149)(71 150)(72 126)(73 127)(74 128)(75 129)

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

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

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

125 conjugacy classes

class 1  2 5A5B5C5D7A7B7C10A10B10C10D25A···25T35A···35L50A···50T175A···175BH
order1255557771010101025···2535···3550···50175···175
size17111122277771···12···27···72···2

125 irreducible representations

dim111111222
type+++
imageC1C2C5C10C25C50D7C5×D7D7×C25
kernelD7×C25C175C5×D7C35D7C7C25C5C1
# reps1144202031260

Matrix representation of D7×C25 in GL3(𝔽701) generated by

62700
010
001
,
100
03131
0559172
,
70000
0172700
0141529
G:=sub<GL(3,GF(701))| [627,0,0,0,1,0,0,0,1],[1,0,0,0,313,559,0,1,172],[700,0,0,0,172,141,0,700,529] >;

D7×C25 in GAP, Magma, Sage, TeX

D_7\times C_{25}
% in TeX

G:=Group("D7xC25");
// GroupNames label

G:=SmallGroup(350,2);
// by ID

G=gap.SmallGroup(350,2);
# by ID

G:=PCGroup([4,-2,-5,-5,-7,45,4803]);
// Polycyclic

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

Export

Subgroup lattice of D7×C25 in TeX

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