direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D60, C20⋊7D6, C4⋊2D30, C30⋊4D4, C6⋊1D20, C10⋊1D12, C12⋊7D10, C60⋊8C22, D30⋊5C22, C30.30C23, C22.10D30, C3⋊2(C2×D20), C5⋊2(C2×D12), (C2×C60)⋊5C2, (C2×C20)⋊3S3, (C2×C12)⋊3D5, (C2×C4)⋊2D15, C15⋊10(C2×D4), (C2×C6).28D10, (C2×C10).28D6, (C22×D15)⋊1C2, C6.30(C22×D5), C2.4(C22×D15), (C2×C30).29C22, C10.30(C22×S3), SmallGroup(240,177)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D60
G = < a,b,c | a2=b60=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 680 in 108 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, D5, C10, C10, C12, D6, C2×C6, C15, C2×D4, C20, D10, C2×C10, D12, C2×C12, C22×S3, D15, C30, C30, D20, C2×C20, C22×D5, C2×D12, C60, D30, D30, C2×C30, C2×D20, D60, C2×C60, C22×D15, C2×D60
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, D15, D20, C22×D5, C2×D12, D30, C2×D20, D60, C22×D15, C2×D60
►On 120 points
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 103)(14 104)(15 105)(16 106)(17 107)(18 108)(19 109)(20 110)(21 111)(22 112)(23 113)(24 114)(25 115)(26 116)(27 117)(28 118)(29 119)(30 120)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 88)(59 89)(60 90)
(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 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(61 120)(62 119)(63 118)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)(79 102)(80 101)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)
G:=sub<Sym(120)| (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)>;
G:=Group( (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,103)(14,104)(15,105)(16,106)(17,107)(18,108)(19,109)(20,110)(21,111)(22,112)(23,113)(24,114)(25,115)(26,116)(27,117)(28,118)(29,119)(30,120)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,81)(52,82)(53,83)(54,84)(55,85)(56,86)(57,87)(58,88)(59,89)(60,90), (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), (1,60)(2,59)(3,58)(4,57)(5,56)(6,55)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(61,120)(62,119)(63,118)(64,117)(65,116)(66,115)(67,114)(68,113)(69,112)(70,111)(71,110)(72,109)(73,108)(74,107)(75,106)(76,105)(77,104)(78,103)(79,102)(80,101)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91) );
G=PermutationGroup([[(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,103),(14,104),(15,105),(16,106),(17,107),(18,108),(19,109),(20,110),(21,111),(22,112),(23,113),(24,114),(25,115),(26,116),(27,117),(28,118),(29,119),(30,120),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(41,71),(42,72),(43,73),(44,74),(45,75),(46,76),(47,77),(48,78),(49,79),(50,80),(51,81),(52,82),(53,83),(54,84),(55,85),(56,86),(57,87),(58,88),(59,89),(60,90)], [(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)], [(1,60),(2,59),(3,58),(4,57),(5,56),(6,55),(7,54),(8,53),(9,52),(10,51),(11,50),(12,49),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,42),(20,41),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(61,120),(62,119),(63,118),(64,117),(65,116),(66,115),(67,114),(68,113),(69,112),(70,111),(71,110),(72,109),(73,108),(74,107),(75,106),(76,105),(77,104),(78,103),(79,102),(80,101),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91)]])
C2×D60 is a maximal subgroup of
C60.29D4 D60⋊12C4 D60⋊15C4 D60⋊9C4 D60⋊8C4 M4(2)⋊D15 D20⋊19D6 C60.38D4 C60.47D4 C60.70D4 D60⋊17C4 D30⋊D4 D60⋊14C4 D30.6D4 C12⋊7D20 C20⋊D12 C12⋊D20 D30⋊2D4 C60⋊6D4 D30⋊5D4 C42⋊6D15 C42⋊7D15 D30⋊16D4 D30⋊9D4 D60⋊11C4 D30.29D4 C4⋊D60 C8⋊D30 C60⋊29D4 C60⋊3D4 C60.23D4 D4⋊D30 C2×D5×D12 C2×S3×D20 D20⋊29D6 C2×D4×D15 D4⋊8D30
C2×D60 is a maximal quotient of
C60⋊8Q8 C42⋊6D15 C42⋊7D15 D30⋊16D4 C22.D60 C4⋊D60 D30⋊6Q8 C40.69D6 C8⋊D30 C8.D30 C60⋊29D4
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D15 | D20 | D30 | D30 | D60 |
| kernel | C2×D60 | D60 | C2×C60 | C22×D15 | C2×C20 | C30 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 |
Matrix representation of C2×D60 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 34 |
| 0 | 0 | 0 | 56 | 7 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 29 | 36 |
| 0 | 0 | 0 | 58 | 32 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[60,0,0,0,0,0,1,60,0,0,0,1,0,0,0,0,0,0,2,56,0,0,0,34,7],[1,0,0,0,0,0,1,0,0,0,0,1,60,0,0,0,0,0,29,58,0,0,0,36,32] >;
C2×D60 in GAP, Magma, Sage, TeX
C_2\times D_{60} % in TeX
G:=Group("C2xD60"); // GroupNames label
G:=SmallGroup(240,177);
// by ID
G=gap.SmallGroup(240,177);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,218,50,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^2=b^60=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations