direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C60⋊C4, D10.11D12, D10.8Dic6, C60⋊7(C2×C4), (C2×C60)⋊3C4, C6⋊1(C4⋊F5), C12⋊7(C2×F5), (C2×C12)⋊5F5, C30⋊2(C4⋊C4), (D5×C12)⋊7C4, D5⋊(C4⋊Dic3), C10⋊(C4⋊Dic3), (C4×D5)⋊4Dic3, (C4×D5).91D6, D5.2(C2×D12), (C6×D5).53D4, (C2×C20)⋊3Dic3, C20⋊2(C2×Dic3), (C6×D5).10Q8, (C6×Dic5)⋊14C4, D5.3(C2×Dic6), C6.36(C22×F5), Dic5⋊7(C2×Dic3), (C2×Dic5)⋊8Dic3, C30.74(C22×C4), (C6×D5).61C23, D10.16(C2×Dic3), (C22×D5).102D6, D10.46(C22×S3), C10.5(C22×Dic3), (D5×C12).116C22, C5⋊(C2×C4⋊Dic3), C3⋊2(C2×C4⋊F5), C4⋊2(C2×C3⋊F5), C15⋊3(C2×C4⋊C4), (C2×C4)⋊3(C3⋊F5), (C2×C4×D5).15S3, (C3×D5)⋊4(C4⋊C4), C2.6(C22×C3⋊F5), (C3×D5).9(C2×D4), (D5×C2×C12).18C2, (C2×C6).47(C2×F5), (C3×D5).5(C2×Q8), (C2×C30).41(C2×C4), (C22×C3⋊F5).5C2, C22.19(C2×C3⋊F5), (C6×D5).59(C2×C4), (C2×C3⋊F5).14C22, (C3×Dic5)⋊26(C2×C4), (D5×C2×C6).144C22, (C2×C10).17(C2×Dic3), SmallGroup(480,1064)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 908 in 184 conjugacy classes, 81 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×6], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×13], C23, D5 [×2], D5 [×2], C10, C10 [×2], Dic3 [×4], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C15, C4⋊C4 [×4], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×4], C2×C10, C2×Dic3 [×8], C2×C12, C2×C12 [×5], C22×C6, C3×D5 [×2], C3×D5 [×2], C30, C30 [×2], C2×C4⋊C4, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×8], C22×D5, C4⋊Dic3 [×4], C22×Dic3 [×2], C22×C12, C3×Dic5 [×2], C60 [×2], C3⋊F5 [×4], C6×D5 [×2], C6×D5 [×4], C2×C30, C4⋊F5 [×4], C2×C4×D5, C22×F5 [×2], C2×C4⋊Dic3, D5×C12 [×4], C6×Dic5, C2×C60, C2×C3⋊F5 [×4], C2×C3⋊F5 [×4], D5×C2×C6, C2×C4⋊F5, C60⋊C4 [×4], D5×C2×C12, C22×C3⋊F5 [×2], C2×C60⋊C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, Dic3 [×4], D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, Dic6 [×2], D12 [×2], C2×Dic3 [×6], C22×S3, C2×C4⋊C4, C2×F5 [×3], C4⋊Dic3 [×4], C2×Dic6, C2×D12, C22×Dic3, C3⋊F5, C4⋊F5 [×2], C22×F5, C2×C4⋊Dic3, C2×C3⋊F5 [×3], C2×C4⋊F5, C60⋊C4 [×2], C22×C3⋊F5, C2×C60⋊C4
Generators and relations
G = < a,b,c | a2=b60=c4=1, ab=ba, ac=ca, cbc-1=b47 >
►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 76)(2 99 50 63)(3 62 39 110)(4 85 28 97)(5 108 17 84)(6 71)(7 94 55 118)(8 117 44 105)(9 80 33 92)(10 103 22 79)(11 66)(12 89 60 113)(13 112 49 100)(14 75 38 87)(15 98 27 74)(16 61)(18 107 54 95)(19 70 43 82)(20 93 32 69)(21 116)(23 102 59 90)(24 65 48 77)(25 88 37 64)(26 111)(29 120 53 72)(30 83 42 119)(31 106)(34 115 58 67)(35 78 47 114)(36 101)(40 73 52 109)(41 96)(45 68 57 104)(46 91)(51 86)(56 81)
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), (1,76)(2,99,50,63)(3,62,39,110)(4,85,28,97)(5,108,17,84)(6,71)(7,94,55,118)(8,117,44,105)(9,80,33,92)(10,103,22,79)(11,66)(12,89,60,113)(13,112,49,100)(14,75,38,87)(15,98,27,74)(16,61)(18,107,54,95)(19,70,43,82)(20,93,32,69)(21,116)(23,102,59,90)(24,65,48,77)(25,88,37,64)(26,111)(29,120,53,72)(30,83,42,119)(31,106)(34,115,58,67)(35,78,47,114)(36,101)(40,73,52,109)(41,96)(45,68,57,104)(46,91)(51,86)(56,81)>;
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), (1,76)(2,99,50,63)(3,62,39,110)(4,85,28,97)(5,108,17,84)(6,71)(7,94,55,118)(8,117,44,105)(9,80,33,92)(10,103,22,79)(11,66)(12,89,60,113)(13,112,49,100)(14,75,38,87)(15,98,27,74)(16,61)(18,107,54,95)(19,70,43,82)(20,93,32,69)(21,116)(23,102,59,90)(24,65,48,77)(25,88,37,64)(26,111)(29,120,53,72)(30,83,42,119)(31,106)(34,115,58,67)(35,78,47,114)(36,101)(40,73,52,109)(41,96)(45,68,57,104)(46,91)(51,86)(56,81) );
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)], [(1,76),(2,99,50,63),(3,62,39,110),(4,85,28,97),(5,108,17,84),(6,71),(7,94,55,118),(8,117,44,105),(9,80,33,92),(10,103,22,79),(11,66),(12,89,60,113),(13,112,49,100),(14,75,38,87),(15,98,27,74),(16,61),(18,107,54,95),(19,70,43,82),(20,93,32,69),(21,116),(23,102,59,90),(24,65,48,77),(25,88,37,64),(26,111),(29,120,53,72),(30,83,42,119),(31,106),(34,115,58,67),(35,78,47,114),(36,101),(40,73,52,109),(41,96),(45,68,57,104),(46,91),(51,86),(56,81)])
Matrix representation ►G ⊆ GL8(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 15 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 38 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 10 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 6 | 33 | 6 |
| 0 | 0 | 0 | 0 | 55 | 55 | 0 | 27 |
| 0 | 0 | 0 | 0 | 34 | 28 | 28 | 34 |
| 0 | 0 | 0 | 0 | 27 | 0 | 55 | 55 |
| 43 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 52 | 18 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 33 | 6 | 0 |
| 0 | 0 | 0 | 0 | 55 | 55 | 0 | 27 |
| 0 | 0 | 0 | 0 | 27 | 0 | 55 | 55 |
| 0 | 0 | 0 | 0 | 0 | 6 | 33 | 6 |
G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[15,38,0,0,0,0,0,0,23,38,0,0,0,0,0,0,0,0,21,29,0,0,0,0,0,0,10,40,0,0,0,0,0,0,0,0,0,55,34,27,0,0,0,0,6,55,28,0,0,0,0,0,33,0,28,55,0,0,0,0,6,27,34,55],[43,52,0,0,0,0,0,0,9,18,0,0,0,0,0,0,0,0,48,4,0,0,0,0,0,0,49,13,0,0,0,0,0,0,0,0,6,55,27,0,0,0,0,0,33,55,0,6,0,0,0,0,6,0,55,33,0,0,0,0,0,27,55,6] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 2 | 10 | 10 | 30 | ··· | 30 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | - | - | + | - | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic3 | Dic3 | D6 | Dic6 | D12 | F5 | C2×F5 | C2×F5 | C3⋊F5 | C4⋊F5 | C2×C3⋊F5 | C2×C3⋊F5 | C60⋊C4 |
| kernel | C2×C60⋊C4 | C60⋊C4 | D5×C2×C12 | C22×C3⋊F5 | D5×C12 | C6×Dic5 | C2×C60 | C2×C4×D5 | C6×D5 | C6×D5 | C4×D5 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | D10 | D10 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{60}\rtimes C_4 % in TeX
G:=Group("C2xC60:C4"); // GroupNames label
G:=SmallGroup(480,1064);
// by ID
G=gap.SmallGroup(480,1064);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,100,2693,14118,2379]);
// Polycyclic
G:=Group<a,b,c|a^2=b^60=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^47>;
// generators/relations