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), C30⋊2(C4⋊C4), (C2×C12)⋊5F5, C12⋊7(C2×F5), (D5×C12)⋊7C4, D5⋊(C4⋊Dic3), C10⋊(C4⋊Dic3), (C4×D5)⋊4Dic3, (C6×D5).53D4, D5.2(C2×D12), (C4×D5).91D6, (C2×C20)⋊3Dic3, C20⋊2(C2×Dic3), (C6×D5).10Q8, (C6×Dic5)⋊14C4, D5.3(C2×Dic6), C6.36(C22×F5), (C2×Dic5)⋊8Dic3, Dic5⋊7(C2×Dic3), 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
Generators and relations for C2×C60⋊C4
G = < a,b,c | a2=b60=c4=1, ab=ba, ac=ca, cbc-1=b47 >
Subgroups: 908 in 184 conjugacy classes, 81 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×Dic3, C2×C12, C2×C12, C22×C6, C3×D5, C3×D5, C30, C30, C2×C4⋊C4, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4⋊Dic3, C22×Dic3, C22×C12, C3×Dic5, C60, C3⋊F5, C6×D5, C6×D5, C2×C30, C4⋊F5, C2×C4×D5, C22×F5, C2×C4⋊Dic3, D5×C12, C6×Dic5, C2×C60, C2×C3⋊F5, C2×C3⋊F5, D5×C2×C6, C2×C4⋊F5, C60⋊C4, D5×C2×C12, C22×C3⋊F5, C2×C60⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, F5, Dic6, D12, C2×Dic3, C22×S3, C2×C4⋊C4, C2×F5, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C3⋊F5, C4⋊F5, C22×F5, C2×C4⋊Dic3, C2×C3⋊F5, C2×C4⋊F5, C60⋊C4, C22×C3⋊F5, C2×C60⋊C4
►On 120 points
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 61)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 71)
(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)
(1 117)(2 80 50 104)(3 103 39 91)(4 66 28 78)(5 89 17 65)(6 112)(7 75 55 99)(8 98 44 86)(9 61 33 73)(10 84 22 120)(11 107)(12 70 60 94)(13 93 49 81)(14 116 38 68)(15 79 27 115)(16 102)(18 88 54 76)(19 111 43 63)(20 74 32 110)(21 97)(23 83 59 71)(24 106 48 118)(25 69 37 105)(26 92)(29 101 53 113)(30 64 42 100)(31 87)(34 96 58 108)(35 119 47 95)(36 82)(40 114 52 90)(41 77)(45 109 57 85)(46 72)(51 67)(56 62)
G:=sub<Sym(120)| (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (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), (1,117)(2,80,50,104)(3,103,39,91)(4,66,28,78)(5,89,17,65)(6,112)(7,75,55,99)(8,98,44,86)(9,61,33,73)(10,84,22,120)(11,107)(12,70,60,94)(13,93,49,81)(14,116,38,68)(15,79,27,115)(16,102)(18,88,54,76)(19,111,43,63)(20,74,32,110)(21,97)(23,83,59,71)(24,106,48,118)(25,69,37,105)(26,92)(29,101,53,113)(30,64,42,100)(31,87)(34,96,58,108)(35,119,47,95)(36,82)(40,114,52,90)(41,77)(45,109,57,85)(46,72)(51,67)(56,62)>;
G:=Group( (1,72)(2,73)(3,74)(4,75)(5,76)(6,77)(7,78)(8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,85)(15,86)(16,87)(17,88)(18,89)(19,90)(20,91)(21,92)(22,93)(23,94)(24,95)(25,96)(26,97)(27,98)(28,99)(29,100)(30,101)(31,102)(32,103)(33,104)(34,105)(35,106)(36,107)(37,108)(38,109)(39,110)(40,111)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,61)(51,62)(52,63)(53,64)(54,65)(55,66)(56,67)(57,68)(58,69)(59,70)(60,71), (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), (1,117)(2,80,50,104)(3,103,39,91)(4,66,28,78)(5,89,17,65)(6,112)(7,75,55,99)(8,98,44,86)(9,61,33,73)(10,84,22,120)(11,107)(12,70,60,94)(13,93,49,81)(14,116,38,68)(15,79,27,115)(16,102)(18,88,54,76)(19,111,43,63)(20,74,32,110)(21,97)(23,83,59,71)(24,106,48,118)(25,69,37,105)(26,92)(29,101,53,113)(30,64,42,100)(31,87)(34,96,58,108)(35,119,47,95)(36,82)(40,114,52,90)(41,77)(45,109,57,85)(46,72)(51,67)(56,62) );
G=PermutationGroup([[(1,72),(2,73),(3,74),(4,75),(5,76),(6,77),(7,78),(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,85),(15,86),(16,87),(17,88),(18,89),(19,90),(20,91),(21,92),(22,93),(23,94),(24,95),(25,96),(26,97),(27,98),(28,99),(29,100),(30,101),(31,102),(32,103),(33,104),(34,105),(35,106),(36,107),(37,108),(38,109),(39,110),(40,111),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,61),(51,62),(52,63),(53,64),(54,65),(55,66),(56,67),(57,68),(58,69),(59,70),(60,71)], [(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)], [(1,117),(2,80,50,104),(3,103,39,91),(4,66,28,78),(5,89,17,65),(6,112),(7,75,55,99),(8,98,44,86),(9,61,33,73),(10,84,22,120),(11,107),(12,70,60,94),(13,93,49,81),(14,116,38,68),(15,79,27,115),(16,102),(18,88,54,76),(19,111,43,63),(20,74,32,110),(21,97),(23,83,59,71),(24,106,48,118),(25,69,37,105),(26,92),(29,101,53,113),(30,64,42,100),(31,87),(34,96,58,108),(35,119,47,95),(36,82),(40,114,52,90),(41,77),(45,109,57,85),(46,72),(51,67),(56,62)]])
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 |
Matrix representation of C2×C60⋊C4 ►in 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] >;
C2×C60⋊C4 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