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G = C3×D75order 450 = 2·32·52

Direct product of C3 and D75

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

Aliases: C3×D75, C751C6, C752S3, C321D25, C15.4D15, C25⋊(C3×S3), C3⋊(C3×D25), (C3×C75)⋊2C2, C5.(C3×D15), (C3×C15).2D5, C15.1(C3×D5), SmallGroup(450,7)

Series: Derived Chief Lower central Upper central

C1C75 — C3×D75
C1C5C25C75C3×C75 — C3×D75
C75 — C3×D75
C1C3

Generators and relations for C3×D75
 G = < a,b,c | a3=b75=c2=1, ab=ba, ac=ca, cbc=b-1 >

75C2
2C3
25S3
75C6
15D5
2C15
25C3×S3
5D15
15C3×D5
3D25
2C75
5C3×D15
3C3×D25

Smallest permutation representation of C3×D75
On 150 points
Generators in S150
(1 51 26)(2 52 27)(3 53 28)(4 54 29)(5 55 30)(6 56 31)(7 57 32)(8 58 33)(9 59 34)(10 60 35)(11 61 36)(12 62 37)(13 63 38)(14 64 39)(15 65 40)(16 66 41)(17 67 42)(18 68 43)(19 69 44)(20 70 45)(21 71 46)(22 72 47)(23 73 48)(24 74 49)(25 75 50)(76 101 126)(77 102 127)(78 103 128)(79 104 129)(80 105 130)(81 106 131)(82 107 132)(83 108 133)(84 109 134)(85 110 135)(86 111 136)(87 112 137)(88 113 138)(89 114 139)(90 115 140)(91 116 141)(92 117 142)(93 118 143)(94 119 144)(95 120 145)(96 121 146)(97 122 147)(98 123 148)(99 124 149)(100 125 150)
(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)
(1 149)(2 148)(3 147)(4 146)(5 145)(6 144)(7 143)(8 142)(9 141)(10 140)(11 139)(12 138)(13 137)(14 136)(15 135)(16 134)(17 133)(18 132)(19 131)(20 130)(21 129)(22 128)(23 127)(24 126)(25 125)(26 124)(27 123)(28 122)(29 121)(30 120)(31 119)(32 118)(33 117)(34 116)(35 115)(36 114)(37 113)(38 112)(39 111)(40 110)(41 109)(42 108)(43 107)(44 106)(45 105)(46 104)(47 103)(48 102)(49 101)(50 100)(51 99)(52 98)(53 97)(54 96)(55 95)(56 94)(57 93)(58 92)(59 91)(60 90)(61 89)(62 88)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(75 150)

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

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

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

117 conjugacy classes

class 1  2 3A3B3C3D3E5A5B6A6B15A···15P25A···25J75A···75CB
order1233333556615···1525···2575···75
size175112222275752···22···22···2

117 irreducible representations

dim11112222222222
type+++++++
imageC1C2C3C6S3D5C3×S3C3×D5D15D25C3×D15C3×D25D75C3×D75
kernelC3×D75C3×C75D75C75C75C3×C15C25C15C15C32C5C3C3C1
# reps112212244108202040

Matrix representation of C3×D75 in GL2(𝔽151) generated by

320
032
,
110
055
,
01
10
G:=sub<GL(2,GF(151))| [32,0,0,32],[11,0,0,55],[0,1,1,0] >;

C3×D75 in GAP, Magma, Sage, TeX

C_3\times D_{75}
% in TeX

G:=Group("C3xD75");
// GroupNames label

G:=SmallGroup(450,7);
// by ID

G=gap.SmallGroup(450,7);
# by ID

G:=PCGroup([5,-2,-3,-3,-5,-5,182,3243,418,9004]);
// Polycyclic

G:=Group<a,b,c|a^3=b^75=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D75 in TeX

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