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G = C3×C40⋊C2order 240 = 24·3·5

Direct product of C3 and C40⋊C2

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

 Derived series C1 — C20 — C3×C40⋊C2
 Chief series C1 — C5 — C10 — C20 — C60 — C3×D20 — C3×C40⋊C2
 Lower central C5 — C10 — C20 — C3×C40⋊C2
 Upper central C1 — C6 — C12 — C24

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

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 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B 6C 6D 8A 8B 10A 10B 12A 12B 12C 12D 15A 15B 15C 15D 20A 20B 20C 20D 24A 24B 24C 24D 30A 30B 30C 30D 40A ··· 40H 60A ··· 60H 120A ··· 120P order 1 2 2 3 3 4 4 5 5 6 6 6 6 8 8 10 10 12 12 12 12 15 15 15 15 20 20 20 20 24 24 24 24 30 30 30 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 20 1 1 2 20 2 2 1 1 20 20 2 2 2 2 2 2 20 20 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

69 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 D4 D5 SD16 D10 C3×D4 C3×D5 D20 C3×SD16 C6×D5 C40⋊C2 C3×D20 C3×C40⋊C2 kernel C3×C40⋊C2 C120 C3×Dic10 C3×D20 C40⋊C2 C40 Dic10 D20 C30 C24 C15 C12 C10 C8 C6 C5 C4 C3 C2 C1 # reps 1 1 1 1 2 2 2 2 1 2 2 2 2 4 4 4 4 8 8 16

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

 7 0 0 7
,
 18 13 15 15
,
 18 4 0 1
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

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