direct product, metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C40⋊C2, C40⋊2C6, C24⋊6D5, C120⋊7C2, C15⋊8SD16, D20.1C6, C6.13D20, C30.23D4, Dic10⋊1C6, C12.52D10, C60.59C22, C8⋊2(C3×D5), C4.8(C6×D5), C5⋊1(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
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
(1 100 52)(2 101 53)(3 102 54)(4 103 55)(5 104 56)(6 105 57)(7 106 58)(8 107 59)(9 108 60)(10 109 61)(11 110 62)(12 111 63)(13 112 64)(14 113 65)(15 114 66)(16 115 67)(17 116 68)(18 117 69)(19 118 70)(20 119 71)(21 120 72)(22 81 73)(23 82 74)(24 83 75)(25 84 76)(26 85 77)(27 86 78)(28 87 79)(29 88 80)(30 89 41)(31 90 42)(32 91 43)(33 92 44)(34 93 45)(35 94 46)(36 95 47)(37 96 48)(38 97 49)(39 98 50)(40 99 51)
(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 43)(42 62)(44 60)(45 79)(46 58)(47 77)(48 56)(49 75)(50 54)(51 73)(53 71)(55 69)(57 67)(59 65)(61 63)(64 80)(66 78)(68 76)(70 74)(81 99)(82 118)(83 97)(84 116)(85 95)(86 114)(87 93)(88 112)(89 91)(90 110)(92 108)(94 106)(96 104)(98 102)(101 119)(103 117)(105 115)(107 113)(109 111)
G:=sub<Sym(120)| (1,100,52)(2,101,53)(3,102,54)(4,103,55)(5,104,56)(6,105,57)(7,106,58)(8,107,59)(9,108,60)(10,109,61)(11,110,62)(12,111,63)(13,112,64)(14,113,65)(15,114,66)(16,115,67)(17,116,68)(18,117,69)(19,118,70)(20,119,71)(21,120,72)(22,81,73)(23,82,74)(24,83,75)(25,84,76)(26,85,77)(27,86,78)(28,87,79)(29,88,80)(30,89,41)(31,90,42)(32,91,43)(33,92,44)(34,93,45)(35,94,46)(36,95,47)(37,96,48)(38,97,49)(39,98,50)(40,99,51), (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)(41,43)(42,62)(44,60)(45,79)(46,58)(47,77)(48,56)(49,75)(50,54)(51,73)(53,71)(55,69)(57,67)(59,65)(61,63)(64,80)(66,78)(68,76)(70,74)(81,99)(82,118)(83,97)(84,116)(85,95)(86,114)(87,93)(88,112)(89,91)(90,110)(92,108)(94,106)(96,104)(98,102)(101,119)(103,117)(105,115)(107,113)(109,111)>;
G:=Group( (1,100,52)(2,101,53)(3,102,54)(4,103,55)(5,104,56)(6,105,57)(7,106,58)(8,107,59)(9,108,60)(10,109,61)(11,110,62)(12,111,63)(13,112,64)(14,113,65)(15,114,66)(16,115,67)(17,116,68)(18,117,69)(19,118,70)(20,119,71)(21,120,72)(22,81,73)(23,82,74)(24,83,75)(25,84,76)(26,85,77)(27,86,78)(28,87,79)(29,88,80)(30,89,41)(31,90,42)(32,91,43)(33,92,44)(34,93,45)(35,94,46)(36,95,47)(37,96,48)(38,97,49)(39,98,50)(40,99,51), (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)(41,43)(42,62)(44,60)(45,79)(46,58)(47,77)(48,56)(49,75)(50,54)(51,73)(53,71)(55,69)(57,67)(59,65)(61,63)(64,80)(66,78)(68,76)(70,74)(81,99)(82,118)(83,97)(84,116)(85,95)(86,114)(87,93)(88,112)(89,91)(90,110)(92,108)(94,106)(96,104)(98,102)(101,119)(103,117)(105,115)(107,113)(109,111) );
G=PermutationGroup([[(1,100,52),(2,101,53),(3,102,54),(4,103,55),(5,104,56),(6,105,57),(7,106,58),(8,107,59),(9,108,60),(10,109,61),(11,110,62),(12,111,63),(13,112,64),(14,113,65),(15,114,66),(16,115,67),(17,116,68),(18,117,69),(19,118,70),(20,119,71),(21,120,72),(22,81,73),(23,82,74),(24,83,75),(25,84,76),(26,85,77),(27,86,78),(28,87,79),(29,88,80),(30,89,41),(31,90,42),(32,91,43),(33,92,44),(34,93,45),(35,94,46),(36,95,47),(37,96,48),(38,97,49),(39,98,50),(40,99,51)], [(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),(41,43),(42,62),(44,60),(45,79),(46,58),(47,77),(48,56),(49,75),(50,54),(51,73),(53,71),(55,69),(57,67),(59,65),(61,63),(64,80),(66,78),(68,76),(70,74),(81,99),(82,118),(83,97),(84,116),(85,95),(86,114),(87,93),(88,112),(89,91),(90,110),(92,108),(94,106),(96,104),(98,102),(101,119),(103,117),(105,115),(107,113),(109,111)]])
C3×C40⋊C2 is a maximal subgroup of
C40⋊1D6 C40⋊14D6 D24⋊6D5 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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