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G = C3×C40⋊C2order 240 = 24·3·5

Direct product of C3 and C40⋊C2

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

Aliases: C3×C40⋊C2, C402C6, C246D5, C1207C2, C158SD16, D20.1C6, C6.13D20, C30.23D4, Dic101C6, C12.52D10, C60.59C22, C82(C3×D5), C4.8(C6×D5), C51(C3×SD16), C20.8(C2×C6), C10.1(C3×D4), C2.3(C3×D20), (C3×D20).3C2, (C3×Dic10)⋊7C2, SmallGroup(240,35)

Series: Derived Chief Lower central Upper central

C1C20 — C3×C40⋊C2
C1C5C10C20C60C3×D20 — C3×C40⋊C2
C5C10C20 — C3×C40⋊C2
C1C6C12C24

Generators and relations for C3×C40⋊C2
 G = < a,b,c | a3=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

20C2
10C22
10C4
20C6
4D5
5Q8
5D4
10C12
10C2×C6
2D10
2Dic5
4C3×D5
5SD16
5C3×Q8
5C3×D4
2C6×D5
2C3×Dic5
5C3×SD16

Smallest permutation representation of C3×C40⋊C2
On 120 points
Generators in S120
(1 82 70)(2 83 71)(3 84 72)(4 85 73)(5 86 74)(6 87 75)(7 88 76)(8 89 77)(9 90 78)(10 91 79)(11 92 80)(12 93 41)(13 94 42)(14 95 43)(15 96 44)(16 97 45)(17 98 46)(18 99 47)(19 100 48)(20 101 49)(21 102 50)(22 103 51)(23 104 52)(24 105 53)(25 106 54)(26 107 55)(27 108 56)(28 109 57)(29 110 58)(30 111 59)(31 112 60)(32 113 61)(33 114 62)(34 115 63)(35 116 64)(36 117 65)(37 118 66)(38 119 67)(39 120 68)(40 81 69)
(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)
(2 20)(3 39)(4 18)(5 37)(6 16)(7 35)(8 14)(9 33)(10 12)(11 31)(13 29)(15 27)(17 25)(19 23)(22 40)(24 38)(26 36)(28 34)(30 32)(41 79)(42 58)(43 77)(44 56)(45 75)(46 54)(47 73)(48 52)(49 71)(51 69)(53 67)(55 65)(57 63)(59 61)(60 80)(62 78)(64 76)(66 74)(68 72)(81 103)(83 101)(84 120)(85 99)(86 118)(87 97)(88 116)(89 95)(90 114)(91 93)(92 112)(94 110)(96 108)(98 106)(100 104)(105 119)(107 117)(109 115)(111 113)

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

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

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

C3×C40⋊C2 is a maximal subgroup of
C401D6  C4014D6  D246D5  D6.1D20  C40.2D6  Dic6.D10  D30.3D4  C3×D5×SD16

69 conjugacy classes

class 1 2A2B3A3B4A4B5A5B6A6B6C6D8A8B10A10B12A12B12C12D15A15B15C15D20A20B20C20D24A24B24C24D30A30B30C30D40A···40H60A···60H120A···120P
order1223344556666881010121212121515151520202020242424243030303040···4060···60120···120
size11201122022112020222222202022222222222222222···22···22···2

69 irreducible representations

dim11111111222222222222
type++++++++
imageC1C2C2C2C3C6C6C6D4D5SD16D10C3×D4C3×D5D20C3×SD16C6×D5C40⋊C2C3×D20C3×C40⋊C2
kernelC3×C40⋊C2C120C3×Dic10C3×D20C40⋊C2C40Dic10D20C30C24C15C12C10C8C6C5C4C3C2C1
# reps111122221222244448816

Matrix representation of C3×C40⋊C2 in GL2(𝔽19) generated by

70
07
,
1813
1515
,
184
01
G:=sub<GL(2,GF(19))| [7,0,0,7],[18,15,13,15],[18,0,4,1] >;

C3×C40⋊C2 in GAP, Magma, Sage, TeX

C_3\times C_{40}\rtimes C_2
% in TeX

G:=Group("C3xC40:C2");
// GroupNames label

G:=SmallGroup(240,35);
// by ID

G=gap.SmallGroup(240,35);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-5,169,79,867,69,6917]);
// 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^19>;
// generators/relations

Export

Subgroup lattice of C3×C40⋊C2 in TeX

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