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G = C3⋊D40order 240 = 24·3·5

The semidirect product of C3 and D40 acting via D40/D20=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C152D8, C32D40, D605C2, D201S3, C6.6D20, C30.2D4, C20.22D6, C12.2D10, C60.8C22, C3⋊C81D5, C51(D4⋊S3), C4.1(S3×D5), (C3×D20)⋊1C2, C10.1(C3⋊D4), C2.4(C3⋊D20), (C5×C3⋊C8)⋊1C2, SmallGroup(240,14)

Series: Derived Chief Lower central Upper central

C1C60 — C3⋊D40
C1C5C15C30C60C3×D20 — C3⋊D40
C15C30C60 — C3⋊D40
C1C2C4

Generators and relations for C3⋊D40
 G = < a,b,c | a3=b40=c2=1, bab-1=cac=a-1, cbc=b-1 >

20C2
60C2
10C22
30C22
20C6
20S3
4D5
12D5
3C8
5D4
15D4
10C2×C6
10D6
2D10
6D10
4D15
4C3×D5
15D8
5C3×D4
5D12
3C40
3D20
2C6×D5
2D30
5D4⋊S3
3D40

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

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

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

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

C3⋊D40 is a maximal subgroup of
S3×D40  C401D6  D40⋊S3  D6.1D20  D2019D6  D20.31D6  D6030C22  D5×D4⋊S3  D15⋊D8  D20.9D6  Dic6⋊D10  D20⋊D6  D60⋊C22  D20.D6  D20.16D6
C3⋊D40 is a maximal quotient of
C3⋊D80  D40.S3  C24.D10  C3⋊Dic40  C6.D40  D6012C4  C60.5Q8

36 conjugacy classes

class 1 2A2B2C 3  4 5A5B6A6B6C8A8B10A10B 12 15A15B20A20B20C20D30A30B40A···40H60A60B60C60D
order1222345566688101012151520202020303040···4060606060
size11206022222202066224442222446···64444

36 irreducible representations

dim11112222222224444
type++++++++++++++++
imageC1C2C2C2S3D4D5D6D8D10C3⋊D4D20D40D4⋊S3S3×D5C3⋊D20C3⋊D40
kernelC3⋊D40C5×C3⋊C8C3×D20D60D20C30C3⋊C8C20C15C12C10C6C3C5C4C2C1
# reps11111121222481224

Matrix representation of C3⋊D40 in GL4(𝔽241) generated by

1000
0100
002401
002400
,
562700
21422800
0001
0010
,
7816300
4416300
0001
0010
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,240,240,0,0,1,0],[56,214,0,0,27,228,0,0,0,0,0,1,0,0,1,0],[78,44,0,0,163,163,0,0,0,0,0,1,0,0,1,0] >;

C3⋊D40 in GAP, Magma, Sage, TeX

C_3\rtimes D_{40}
% in TeX

G:=Group("C3:D40");
// GroupNames label

G:=SmallGroup(240,14);
// by ID

G=gap.SmallGroup(240,14);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,73,79,218,50,490,6917]);
// Polycyclic

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

Export

Subgroup lattice of C3⋊D40 in TeX

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