direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20⋊D6, D30⋊15D4, D20⋊23D6, C60⋊5C23, D12⋊23D10, C30.17C24, Dic15⋊7C23, D30.37C23, C6⋊2(D4×D5), C30⋊3(C2×D4), C10⋊2(S3×D4), (C2×C20)⋊5D6, D15⋊1(C2×D4), (C2×C12)⋊5D10, C15⋊3(C22×D4), (C2×D20)⋊13S3, (C2×D12)⋊13D5, (C6×D20)⋊13C2, C20⋊3(C22×S3), D6⋊2(C22×D5), (C6×D5)⋊2C23, C12⋊3(C22×D5), (C10×D12)⋊13C2, (S3×C10)⋊2C23, (C2×C60)⋊16C22, (C22×D5)⋊10D6, D10⋊2(C22×S3), (C22×S3)⋊9D10, C6.17(C23×D5), (C3×D20)⋊30C22, (C4×D15)⋊23C22, (C5×D12)⋊30C22, C15⋊D4⋊10C22, C10.17(S3×C23), (C2×C30).236C23, (C2×Dic15)⋊33C22, (C22×D15).121C22, C3⋊2(C2×D4×D5), C5⋊2(C2×S3×D4), C4⋊4(C2×S3×D5), (C2×C4×D15)⋊26C2, (C2×C4)⋊14(S3×D5), (C22×S3×D5)⋊7C2, (D5×C2×C6)⋊5C22, (S3×C2×C10)⋊5C22, (C2×S3×D5)⋊11C22, (C2×C15⋊D4)⋊19C2, C2.20(C22×S3×D5), C22.105(C2×S3×D5), (C2×C6).246(C22×D5), (C2×C10).246(C22×S3), SmallGroup(480,1089)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 2876 in 472 conjugacy classes, 124 normal (24 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×4], D6 [×26], C2×C6, C2×C6 [×8], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], C20 [×2], D10 [×4], D10 [×26], C2×C10, C2×C10 [×8], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3 [×2], C22×S3 [×17], C22×C6 [×2], C5×S3 [×4], C3×D5 [×4], D15 [×4], C30, C30 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5 [×2], C22×D5 [×17], C22×C10 [×2], S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], Dic15 [×2], C60 [×2], S3×D5 [×16], C6×D5 [×4], C6×D5 [×4], S3×C10 [×4], S3×C10 [×4], D30 [×6], C2×C30, C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C2×S3×D4, C15⋊D4 [×8], C3×D20 [×4], C5×D12 [×4], C4×D15 [×4], C2×Dic15, C2×C60, C2×S3×D5 [×8], C2×S3×D5 [×8], D5×C2×C6 [×2], S3×C2×C10 [×2], C22×D15, C2×D4×D5, C20⋊D6 [×8], C2×C15⋊D4 [×2], C6×D20, C10×D12, C2×C4×D15, C22×S3×D5 [×2], C2×C20⋊D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], C22×D4, C22×D5 [×7], S3×D4 [×2], S3×C23, S3×D5, D4×D5 [×2], C23×D5, C2×S3×D4, C2×S3×D5 [×3], C2×D4×D5, C20⋊D6 [×2], C22×S3×D5, C2×C20⋊D6
Generators and relations
G = < a,b,c,d | a2=b20=c6=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=b9, dcd=c-1 >
►On 120 points
(1 91)(2 92)(3 93)(4 94)(5 95)(6 96)(7 97)(8 98)(9 99)(10 100)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 113)(32 114)(33 115)(34 116)(35 117)(36 118)(37 119)(38 120)(39 101)(40 102)(41 80)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)(49 68)(50 69)(51 70)(52 71)(53 72)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)
(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)
(1 108 47)(2 107 48 20 109 46)(3 106 49 19 110 45)(4 105 50 18 111 44)(5 104 51 17 112 43)(6 103 52 16 113 42)(7 102 53 15 114 41)(8 101 54 14 115 60)(9 120 55 13 116 59)(10 119 56 12 117 58)(11 118 57)(21 71 86 31 61 96)(22 70 87 30 62 95)(23 69 88 29 63 94)(24 68 89 28 64 93)(25 67 90 27 65 92)(26 66 91)(32 80 97 40 72 85)(33 79 98 39 73 84)(34 78 99 38 74 83)(35 77 100 37 75 82)(36 76 81)
(1 76)(2 65)(3 74)(4 63)(5 72)(6 61)(7 70)(8 79)(9 68)(10 77)(11 66)(12 75)(13 64)(14 73)(15 62)(16 71)(17 80)(18 69)(19 78)(20 67)(21 113)(22 102)(23 111)(24 120)(25 109)(26 118)(27 107)(28 116)(29 105)(30 114)(31 103)(32 112)(33 101)(34 110)(35 119)(36 108)(37 117)(38 106)(39 115)(40 104)(41 87)(42 96)(43 85)(44 94)(45 83)(46 92)(47 81)(48 90)(49 99)(50 88)(51 97)(52 86)(53 95)(54 84)(55 93)(56 82)(57 91)(58 100)(59 89)(60 98)
G:=sub<Sym(120)| (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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), (1,108,47)(2,107,48,20,109,46)(3,106,49,19,110,45)(4,105,50,18,111,44)(5,104,51,17,112,43)(6,103,52,16,113,42)(7,102,53,15,114,41)(8,101,54,14,115,60)(9,120,55,13,116,59)(10,119,56,12,117,58)(11,118,57)(21,71,86,31,61,96)(22,70,87,30,62,95)(23,69,88,29,63,94)(24,68,89,28,64,93)(25,67,90,27,65,92)(26,66,91)(32,80,97,40,72,85)(33,79,98,39,73,84)(34,78,99,38,74,83)(35,77,100,37,75,82)(36,76,81), (1,76)(2,65)(3,74)(4,63)(5,72)(6,61)(7,70)(8,79)(9,68)(10,77)(11,66)(12,75)(13,64)(14,73)(15,62)(16,71)(17,80)(18,69)(19,78)(20,67)(21,113)(22,102)(23,111)(24,120)(25,109)(26,118)(27,107)(28,116)(29,105)(30,114)(31,103)(32,112)(33,101)(34,110)(35,119)(36,108)(37,117)(38,106)(39,115)(40,104)(41,87)(42,96)(43,85)(44,94)(45,83)(46,92)(47,81)(48,90)(49,99)(50,88)(51,97)(52,86)(53,95)(54,84)(55,93)(56,82)(57,91)(58,100)(59,89)(60,98)>;
G:=Group( (1,91)(2,92)(3,93)(4,94)(5,95)(6,96)(7,97)(8,98)(9,99)(10,100)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,103)(22,104)(23,105)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,113)(32,114)(33,115)(34,116)(35,117)(36,118)(37,119)(38,120)(39,101)(40,102)(41,80)(42,61)(43,62)(44,63)(45,64)(46,65)(47,66)(48,67)(49,68)(50,69)(51,70)(52,71)(53,72)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79), (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), (1,108,47)(2,107,48,20,109,46)(3,106,49,19,110,45)(4,105,50,18,111,44)(5,104,51,17,112,43)(6,103,52,16,113,42)(7,102,53,15,114,41)(8,101,54,14,115,60)(9,120,55,13,116,59)(10,119,56,12,117,58)(11,118,57)(21,71,86,31,61,96)(22,70,87,30,62,95)(23,69,88,29,63,94)(24,68,89,28,64,93)(25,67,90,27,65,92)(26,66,91)(32,80,97,40,72,85)(33,79,98,39,73,84)(34,78,99,38,74,83)(35,77,100,37,75,82)(36,76,81), (1,76)(2,65)(3,74)(4,63)(5,72)(6,61)(7,70)(8,79)(9,68)(10,77)(11,66)(12,75)(13,64)(14,73)(15,62)(16,71)(17,80)(18,69)(19,78)(20,67)(21,113)(22,102)(23,111)(24,120)(25,109)(26,118)(27,107)(28,116)(29,105)(30,114)(31,103)(32,112)(33,101)(34,110)(35,119)(36,108)(37,117)(38,106)(39,115)(40,104)(41,87)(42,96)(43,85)(44,94)(45,83)(46,92)(47,81)(48,90)(49,99)(50,88)(51,97)(52,86)(53,95)(54,84)(55,93)(56,82)(57,91)(58,100)(59,89)(60,98) );
G=PermutationGroup([(1,91),(2,92),(3,93),(4,94),(5,95),(6,96),(7,97),(8,98),(9,99),(10,100),(11,81),(12,82),(13,83),(14,84),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,103),(22,104),(23,105),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,113),(32,114),(33,115),(34,116),(35,117),(36,118),(37,119),(38,120),(39,101),(40,102),(41,80),(42,61),(43,62),(44,63),(45,64),(46,65),(47,66),(48,67),(49,68),(50,69),(51,70),(52,71),(53,72),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79)], [(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)], [(1,108,47),(2,107,48,20,109,46),(3,106,49,19,110,45),(4,105,50,18,111,44),(5,104,51,17,112,43),(6,103,52,16,113,42),(7,102,53,15,114,41),(8,101,54,14,115,60),(9,120,55,13,116,59),(10,119,56,12,117,58),(11,118,57),(21,71,86,31,61,96),(22,70,87,30,62,95),(23,69,88,29,63,94),(24,68,89,28,64,93),(25,67,90,27,65,92),(26,66,91),(32,80,97,40,72,85),(33,79,98,39,73,84),(34,78,99,38,74,83),(35,77,100,37,75,82),(36,76,81)], [(1,76),(2,65),(3,74),(4,63),(5,72),(6,61),(7,70),(8,79),(9,68),(10,77),(11,66),(12,75),(13,64),(14,73),(15,62),(16,71),(17,80),(18,69),(19,78),(20,67),(21,113),(22,102),(23,111),(24,120),(25,109),(26,118),(27,107),(28,116),(29,105),(30,114),(31,103),(32,112),(33,101),(34,110),(35,119),(36,108),(37,117),(38,106),(39,115),(40,104),(41,87),(42,96),(43,85),(44,94),(45,83),(46,92),(47,81),(48,90),(49,99),(50,88),(51,97),(52,86),(53,95),(54,84),(55,93),(56,82),(57,91),(58,100),(59,89),(60,98)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 18 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 43 | 60 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,60,18],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,43,0,0,0,0,0,60],[0,60,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,18,0,0,0,0,0,1] >;
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | C2×S3×D5 | C20⋊D6 |
| kernel | C2×C20⋊D6 | C20⋊D6 | C2×C15⋊D4 | C6×D20 | C10×D12 | C2×C4×D15 | C22×S3×D5 | C2×D20 | D30 | C2×D12 | D20 | C2×C20 | C22×D5 | D12 | C2×C12 | C22×S3 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 4 | 1 | 2 | 8 | 2 | 4 | 2 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_{20}\rtimes D_6 % in TeX
G:=Group("C2xC20:D6"); // GroupNames label
G:=SmallGroup(480,1089);
// by ID
G=gap.SmallGroup(480,1089);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=b^9,d*c*d=c^-1>;
// generators/relations