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G = C6×D25order 300 = 22·3·52

Direct product of C6 and D25

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

Aliases: C6×D25, C50⋊C6, C1502C2, C753C22, C30.5D5, C15.3D10, C25⋊(C2×C6), C5.(C6×D5), C10.2(C3×D5), SmallGroup(300,9)

Series: Derived Chief Lower central Upper central

C1C25 — C6×D25
C1C5C25C75C3×D25 — C6×D25
C25 — C6×D25
C1C6

Generators and relations for C6×D25
 G = < a,b,c | a6=b25=c2=1, ab=ba, ac=ca, cbc=b-1 >

25C2
25C2
25C22
25C6
25C6
5D5
5D5
25C2×C6
5D10
5C3×D5
5C3×D5
5C6×D5

Smallest permutation representation of C6×D25
On 150 points
Generators in S150
(1 125 57 84 35 142)(2 101 58 85 36 143)(3 102 59 86 37 144)(4 103 60 87 38 145)(5 104 61 88 39 146)(6 105 62 89 40 147)(7 106 63 90 41 148)(8 107 64 91 42 149)(9 108 65 92 43 150)(10 109 66 93 44 126)(11 110 67 94 45 127)(12 111 68 95 46 128)(13 112 69 96 47 129)(14 113 70 97 48 130)(15 114 71 98 49 131)(16 115 72 99 50 132)(17 116 73 100 26 133)(18 117 74 76 27 134)(19 118 75 77 28 135)(20 119 51 78 29 136)(21 120 52 79 30 137)(22 121 53 80 31 138)(23 122 54 81 32 139)(24 123 55 82 33 140)(25 124 56 83 34 141)
(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)
(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 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(44 50)(45 49)(46 48)(51 62)(52 61)(53 60)(54 59)(55 58)(56 57)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(76 91)(77 90)(78 89)(79 88)(80 87)(81 86)(82 85)(83 84)(92 100)(93 99)(94 98)(95 97)(101 123)(102 122)(103 121)(104 120)(105 119)(106 118)(107 117)(108 116)(109 115)(110 114)(111 113)(124 125)(126 132)(127 131)(128 130)(133 150)(134 149)(135 148)(136 147)(137 146)(138 145)(139 144)(140 143)(141 142)

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

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

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

84 conjugacy classes

class 1 2A2B2C3A3B5A5B6A6B6C6D6E6F10A10B15A15B15C15D25A···25J30A30B30C30D50A···50J75A···75T150A···150T
order1222335566666610101515151525···253030303050···5075···75150···150
size112525112211252525252222222···222222···22···22···2

84 irreducible representations

dim11111122222222
type+++++++
imageC1C2C2C3C6C6D5D10C3×D5D25C6×D5D50C3×D25C6×D25
kernelC6×D25C3×D25C150D50D25C50C30C15C10C6C5C3C2C1
# reps121242224104102020

Matrix representation of C6×D25 in GL3(𝔽151) generated by

15000
0320
0032
,
100
03680
07111
,
15000
01174
080140
G:=sub<GL(3,GF(151))| [150,0,0,0,32,0,0,0,32],[1,0,0,0,36,71,0,80,11],[150,0,0,0,11,80,0,74,140] >;

C6×D25 in GAP, Magma, Sage, TeX

C_6\times D_{25}
% in TeX

G:=Group("C6xD25");
// GroupNames label

G:=SmallGroup(300,9);
// by ID

G=gap.SmallGroup(300,9);
# by ID

G:=PCGroup([5,-2,-2,-3,-5,-5,2163,418,6004]);
// Polycyclic

G:=Group<a,b,c|a^6=b^25=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D25 in TeX

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