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G = C40⋊S3order 240 = 24·3·5

4th semidirect product of C40 and S3 acting via S3/C3=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C404S3, C245D5, C83D15, C1205C2, D30.3C4, C20.48D6, C4.13D30, C12.49D10, C1510M4(2), C60.55C22, Dic15.3C4, C153C84C2, C6.6(C4×D5), C54(C8⋊S3), C32(C8⋊D5), C2.3(C4×D15), C10.13(C4×S3), C30.36(C2×C4), (C4×D15).2C2, SmallGroup(240,66)

Series: Derived Chief Lower central Upper central

C1C30 — C40⋊S3
C1C5C15C30C60C4×D15 — C40⋊S3
C15C30 — C40⋊S3
C1C4C8

Generators and relations for C40⋊S3
 G = < a,b,c | a40=b3=c2=1, ab=ba, cac=a29, cbc=b-1 >

30C2
15C22
15C4
10S3
6D5
15C2×C4
15C8
5Dic3
5D6
3Dic5
3D10
2D15
15M4(2)
5C3⋊C8
5C4×S3
3C4×D5
3C52C8
5C8⋊S3
3C8⋊D5

Smallest permutation representation of C40⋊S3
On 120 points
Generators in S120
(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 97 55)(2 98 56)(3 99 57)(4 100 58)(5 101 59)(6 102 60)(7 103 61)(8 104 62)(9 105 63)(10 106 64)(11 107 65)(12 108 66)(13 109 67)(14 110 68)(15 111 69)(16 112 70)(17 113 71)(18 114 72)(19 115 73)(20 116 74)(21 117 75)(22 118 76)(23 119 77)(24 120 78)(25 81 79)(26 82 80)(27 83 41)(28 84 42)(29 85 43)(30 86 44)(31 87 45)(32 88 46)(33 89 47)(34 90 48)(35 91 49)(36 92 50)(37 93 51)(38 94 52)(39 95 53)(40 96 54)
(2 30)(3 19)(4 8)(5 37)(6 26)(7 15)(9 33)(10 22)(12 40)(13 29)(14 18)(16 36)(17 25)(20 32)(23 39)(24 28)(27 35)(34 38)(41 91)(42 120)(43 109)(44 98)(45 87)(46 116)(47 105)(48 94)(49 83)(50 112)(51 101)(52 90)(53 119)(54 108)(55 97)(56 86)(57 115)(58 104)(59 93)(60 82)(61 111)(62 100)(63 89)(64 118)(65 107)(66 96)(67 85)(68 114)(69 103)(70 92)(71 81)(72 110)(73 99)(74 88)(75 117)(76 106)(77 95)(78 84)(79 113)(80 102)

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

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

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

C40⋊S3 is a maximal subgroup of
D5×C8⋊S3  S3×C8⋊D5  D246D5  C408D6  C40.34D6  C40.55D6  D30.3D4  D30.4D4  D60.6C4  M4(2)×D15  D60.3C4  D8⋊D15  Q83D30  SD16⋊D15  Q16⋊D15
C40⋊S3 is a maximal quotient of
C60.26Q8  C12013C4  D303C8

66 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B 6 8A8B8C8D10A10B12A12B15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order12234445568888101012121515151520202020242424243030303040···4060···60120···120
size113021130222223030222222222222222222222···22···22···2

66 irreducible representations

dim1111112222222222222
type++++++++++
imageC1C2C2C2C4C4S3D5D6M4(2)D10C4×S3D15C4×D5C8⋊S3D30C8⋊D5C4×D15C40⋊S3
kernelC40⋊S3C153C8C120C4×D15Dic15D30C40C24C20C15C12C10C8C6C5C4C3C2C1
# reps11112212122244448816

Matrix representation of C40⋊S3 in GL2(𝔽29) generated by

39
820
,
209
88
,
72
522
G:=sub<GL(2,GF(29))| [3,8,9,20],[20,8,9,8],[7,5,2,22] >;

C40⋊S3 in GAP, Magma, Sage, TeX

C_{40}\rtimes S_3
% in TeX

G:=Group("C40:S3");
// GroupNames label

G:=SmallGroup(240,66);
// by ID

G=gap.SmallGroup(240,66);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,50,964,6917]);
// Polycyclic

G:=Group<a,b,c|a^40=b^3=c^2=1,a*b=b*a,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C40⋊S3 in TeX

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