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G = C3×D69order 414 = 2·32·23

Direct product of C3 and D69

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

Aliases: C3×D69, C691C6, C692S3, C321D23, C23⋊(C3×S3), C3⋊(C3×D23), (C3×C69)⋊2C2, SmallGroup(414,7)

Series: Derived Chief Lower central Upper central

C1C69 — C3×D69
C1C23C69C3×C69 — C3×D69
C69 — C3×D69
C1C3

Generators and relations for C3×D69
 G = < a,b,c | a3=b69=c2=1, ab=ba, ac=ca, cbc=b-1 >

69C2
2C3
23S3
69C6
3D23
2C69
23C3×S3
3C3×D23

Smallest permutation representation of C3×D69
On 138 points
Generators in S138
(1 24 47)(2 25 48)(3 26 49)(4 27 50)(5 28 51)(6 29 52)(7 30 53)(8 31 54)(9 32 55)(10 33 56)(11 34 57)(12 35 58)(13 36 59)(14 37 60)(15 38 61)(16 39 62)(17 40 63)(18 41 64)(19 42 65)(20 43 66)(21 44 67)(22 45 68)(23 46 69)(70 116 93)(71 117 94)(72 118 95)(73 119 96)(74 120 97)(75 121 98)(76 122 99)(77 123 100)(78 124 101)(79 125 102)(80 126 103)(81 127 104)(82 128 105)(83 129 106)(84 130 107)(85 131 108)(86 132 109)(87 133 110)(88 134 111)(89 135 112)(90 136 113)(91 137 114)(92 138 115)
(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)
(1 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 84)(11 83)(12 82)(13 81)(14 80)(15 79)(16 78)(17 77)(18 76)(19 75)(20 74)(21 73)(22 72)(23 71)(24 70)(25 138)(26 137)(27 136)(28 135)(29 134)(30 133)(31 132)(32 131)(33 130)(34 129)(35 128)(36 127)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 108)(56 107)(57 106)(58 105)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)

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

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

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

108 conjugacy classes

class 1  2 3A3B3C3D3E6A6B23A···23K69A···69CJ
order12333336623···2369···69
size1691122269692···22···2

108 irreducible representations

dim1111222222
type+++++
imageC1C2C3C6S3C3×S3D23C3×D23D69C3×D69
kernelC3×D69C3×C69D69C69C69C23C32C3C3C1
# reps11221211222244

Matrix representation of C3×D69 in GL2(𝔽139) generated by

420
042
,
350
04
,
04
350
G:=sub<GL(2,GF(139))| [42,0,0,42],[35,0,0,4],[0,35,4,0] >;

C3×D69 in GAP, Magma, Sage, TeX

C_3\times D_{69}
% in TeX

G:=Group("C3xD69");
// GroupNames label

G:=SmallGroup(414,7);
// by ID

G=gap.SmallGroup(414,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-23,146,6339]);
// Polycyclic

G:=Group<a,b,c|a^3=b^69=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D69 in TeX

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