metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊6C2, D6.6D10, C20.28D6, Dic10⋊5S3, C12.17D10, C30.6C23, Dic5.3D6, C60.14C22, D30.1C22, Dic3.11D10, (C4×S3)⋊2D5, C4.7(S3×D5), (S3×C20)⋊2C2, C3⋊1(C4○D20), C15⋊4(C4○D4), C5⋊D12⋊2C2, D30.C2⋊1C2, C5⋊1(Q8⋊3S3), C6.6(C22×D5), (C3×Dic10)⋊3C2, C10.6(C22×S3), (S3×C10).7C22, (C5×Dic3).9C22, (C3×Dic5).3C22, C2.10(C2×S3×D5), SmallGroup(240,130)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊C2
G = < a,b,c | a60=b2=c2=1, bab=a-1, cac=a41, cbc=a10b >
Subgroups: 376 in 80 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, D6, C15, C4○D4, Dic5, C20, C20, D10, C2×C10, C4×S3, C4×S3, D12, C3×Q8, C5×S3, D15, C30, Dic10, C4×D5, D20, C5⋊D4, C2×C20, Q8⋊3S3, C5×Dic3, C3×Dic5, C60, S3×C10, D30, C4○D20, D30.C2, C5⋊D12, C3×Dic10, S3×C20, D60, D60⋊C2
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, D10, C22×S3, C22×D5, Q8⋊3S3, S3×D5, C4○D20, C2×S3×D5, D60⋊C2
►On 120 points
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(61 83)(62 82)(63 81)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 113)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)(99 105)(100 104)(101 103)
(1 110)(2 91)(3 72)(4 113)(5 94)(6 75)(7 116)(8 97)(9 78)(10 119)(11 100)(12 81)(13 62)(14 103)(15 84)(16 65)(17 106)(18 87)(19 68)(20 109)(21 90)(22 71)(23 112)(24 93)(25 74)(26 115)(27 96)(28 77)(29 118)(30 99)(31 80)(32 61)(33 102)(34 83)(35 64)(36 105)(37 86)(38 67)(39 108)(40 89)(41 70)(42 111)(43 92)(44 73)(45 114)(46 95)(47 76)(48 117)(49 98)(50 79)(51 120)(52 101)(53 82)(54 63)(55 104)(56 85)(57 66)(58 107)(59 88)(60 69)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,110)(2,91)(3,72)(4,113)(5,94)(6,75)(7,116)(8,97)(9,78)(10,119)(11,100)(12,81)(13,62)(14,103)(15,84)(16,65)(17,106)(18,87)(19,68)(20,109)(21,90)(22,71)(23,112)(24,93)(25,74)(26,115)(27,96)(28,77)(29,118)(30,99)(31,80)(32,61)(33,102)(34,83)(35,64)(36,105)(37,86)(38,67)(39,108)(40,89)(41,70)(42,111)(43,92)(44,73)(45,114)(46,95)(47,76)(48,117)(49,98)(50,79)(51,120)(52,101)(53,82)(54,63)(55,104)(56,85)(57,66)(58,107)(59,88)(60,69)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(61,83)(62,82)(63,81)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)(84,120)(85,119)(86,118)(87,117)(88,116)(89,115)(90,114)(91,113)(92,112)(93,111)(94,110)(95,109)(96,108)(97,107)(98,106)(99,105)(100,104)(101,103), (1,110)(2,91)(3,72)(4,113)(5,94)(6,75)(7,116)(8,97)(9,78)(10,119)(11,100)(12,81)(13,62)(14,103)(15,84)(16,65)(17,106)(18,87)(19,68)(20,109)(21,90)(22,71)(23,112)(24,93)(25,74)(26,115)(27,96)(28,77)(29,118)(30,99)(31,80)(32,61)(33,102)(34,83)(35,64)(36,105)(37,86)(38,67)(39,108)(40,89)(41,70)(42,111)(43,92)(44,73)(45,114)(46,95)(47,76)(48,117)(49,98)(50,79)(51,120)(52,101)(53,82)(54,63)(55,104)(56,85)(57,66)(58,107)(59,88)(60,69) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,60),(17,59),(18,58),(19,57),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(61,83),(62,82),(63,81),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73),(84,120),(85,119),(86,118),(87,117),(88,116),(89,115),(90,114),(91,113),(92,112),(93,111),(94,110),(95,109),(96,108),(97,107),(98,106),(99,105),(100,104),(101,103)], [(1,110),(2,91),(3,72),(4,113),(5,94),(6,75),(7,116),(8,97),(9,78),(10,119),(11,100),(12,81),(13,62),(14,103),(15,84),(16,65),(17,106),(18,87),(19,68),(20,109),(21,90),(22,71),(23,112),(24,93),(25,74),(26,115),(27,96),(28,77),(29,118),(30,99),(31,80),(32,61),(33,102),(34,83),(35,64),(36,105),(37,86),(38,67),(39,108),(40,89),(41,70),(42,111),(43,92),(44,73),(45,114),(46,95),(47,76),(48,117),(49,98),(50,79),(51,120),(52,101),(53,82),(54,63),(55,104),(56,85),(57,66),(58,107),(59,88),(60,69)]])
D60⋊C2 is a maximal subgroup of
C40⋊1D6 Dic20⋊S3 D6.1D20 D120⋊5C2 C60.19C23 D12.9D10 Dic10.27D6 C60.44C23 C30.C24 S3×C4○D20 D20⋊29D6 D30.C23 D12⋊14D10 C30.33C24 D5×Q8⋊3S3
D60⋊C2 is a maximal quotient of
(C2×C20).D6 Dic3×Dic10 Dic5.1Dic6 (S3×C20)⋊5C4 C60⋊5C4⋊C2 C4⋊Dic5⋊S3 D6⋊Dic5⋊C2 C60.46D4 C60.47D4 C10.D4⋊S3 C60.6Q8 D30.23(C2×C4) D30⋊Q8 D30⋊D4 D60⋊14C4 D30.6D4 D30.7D4 C15⋊22(C4×D4) C60⋊6D4
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 6 | 30 | 30 | 2 | 2 | 3 | 3 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C4○D20 | Q8⋊3S3 | S3×D5 | C2×S3×D5 | D60⋊C2 |
| kernel | D60⋊C2 | D30.C2 | C5⋊D12 | C3×Dic10 | S3×C20 | D60 | Dic10 | C4×S3 | Dic5 | C20 | C15 | Dic3 | C12 | D6 | C3 | C5 | C4 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of D60⋊C2 ►in GL4(𝔽61) generated by
| 29 | 2 | 0 | 0 |
| 59 | 2 | 0 | 0 |
| 0 | 0 | 2 | 5 |
| 0 | 0 | 36 | 60 |
| 1 | 44 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 59 | 56 |
| 0 | 0 | 25 | 2 |
| 14 | 45 | 0 | 0 |
| 16 | 47 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 36 | 60 |
G:=sub<GL(4,GF(61))| [29,59,0,0,2,2,0,0,0,0,2,36,0,0,5,60],[1,0,0,0,44,60,0,0,0,0,59,25,0,0,56,2],[14,16,0,0,45,47,0,0,0,0,1,36,0,0,0,60] >;
D60⋊C2 in GAP, Magma, Sage, TeX
D_{60}\rtimes C_2 % in TeX
G:=Group("D60:C2"); // GroupNames label
G:=SmallGroup(240,130);
// by ID
G=gap.SmallGroup(240,130);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,55,218,50,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^2=1,b*a*b=a^-1,c*a*c=a^41,c*b*c=a^10*b>;
// generators/relations