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G = C3×D40order 240 = 24·3·5

Direct product of C3 and D40

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

Aliases: C3×D40, C155D8, C401C6, C243D5, C1203C2, D201C6, C6.14D20, C30.24D4, C12.53D10, C60.60C22, C51(C3×D8), C81(C3×D5), C4.9(C6×D5), (C3×D20)⋊7C2, C20.9(C2×C6), C10.2(C3×D4), C2.4(C3×D20), SmallGroup(240,36)

Series: Derived Chief Lower central Upper central

C1C20 — C3×D40
C1C5C10C20C60C3×D20 — C3×D40
C5C10C20 — C3×D40
C1C6C12C24

Generators and relations for C3×D40
 G = < a,b,c | a3=b40=c2=1, ab=ba, ac=ca, cbc=b-1 >

20C2
20C2
10C22
10C22
20C6
20C6
4D5
4D5
5D4
5D4
10C2×C6
10C2×C6
2D10
2D10
4C3×D5
4C3×D5
5D8
5C3×D4
5C3×D4
2C6×D5
2C6×D5
5C3×D8

Smallest permutation representation of C3×D40
On 120 points
Generators in S120
(1 106 79)(2 107 80)(3 108 41)(4 109 42)(5 110 43)(6 111 44)(7 112 45)(8 113 46)(9 114 47)(10 115 48)(11 116 49)(12 117 50)(13 118 51)(14 119 52)(15 120 53)(16 81 54)(17 82 55)(18 83 56)(19 84 57)(20 85 58)(21 86 59)(22 87 60)(23 88 61)(24 89 62)(25 90 63)(26 91 64)(27 92 65)(28 93 66)(29 94 67)(30 95 68)(31 96 69)(32 97 70)(33 98 71)(34 99 72)(35 100 73)(36 101 74)(37 102 75)(38 103 76)(39 104 77)(40 105 78)
(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)
(1 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)(41 76)(42 75)(43 74)(44 73)(45 72)(46 71)(47 70)(48 69)(49 68)(50 67)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 60)(58 59)(77 80)(78 79)(81 90)(82 89)(83 88)(84 87)(85 86)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)

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

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

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

C3×D40 is a maximal subgroup of
C15⋊D16  C3⋊D80  C40.D6  D40.S3  D40⋊S3  C405D6  C408D6  D407S3  D405S3  C3×D5×D8

69 conjugacy classes

class 1 2A2B2C3A3B 4 5A5B6A6B6C6D6E6F8A8B10A10B12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12223345566666688101012121515151520202020242424243030303040···4060···60120···120
size11202011222112020202022222222222222222222222···22···22···2

69 irreducible representations

dim111111222222222222
type+++++++++
imageC1C2C2C3C6C6D4D5D8D10C3×D4C3×D5D20C3×D8C6×D5D40C3×D20C3×D40
kernelC3×D40C120C3×D20D40C40D20C30C24C15C12C10C8C6C5C4C3C2C1
# reps1122241222244448816

Matrix representation of C3×D40 in GL2(𝔽241) generated by

150
015
,
227194
4720
,
227194
19914
G:=sub<GL(2,GF(241))| [15,0,0,15],[227,47,194,20],[227,199,194,14] >;

C3×D40 in GAP, Magma, Sage, TeX

C_3\times D_{40}
% in TeX

G:=Group("C3xD40");
// GroupNames label

G:=SmallGroup(240,36);
// by ID

G=gap.SmallGroup(240,36);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,169,223,867,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^3=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3×D40 in TeX

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