metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C6).2D20, C15⋊7(C23⋊C4), (C6×Dic5)⋊1C4, (C2×C30).36D4, C23.8(S3×D5), C6.D4⋊1D5, (C2×Dic5)⋊1Dic3, (C22×D5)⋊3Dic3, (C22×C10).27D6, (C22×C6).12D10, C5⋊3(C23.7D6), C30.38D4⋊14C2, C22.5(D5×Dic3), C30.66(C22⋊C4), C22.8(C15⋊D4), C22.8(C3⋊D20), C3⋊3(C23.1D10), C6.32(D10⋊C4), (C22×C30).26C22, C10.22(C6.D4), C2.11(D10⋊Dic3), (D5×C2×C6)⋊1C4, (C2×C6).49(C4×D5), (C6×C5⋊D4).1C2, (C2×C5⋊D4).1S3, (C2×C30).90(C2×C4), (C2×C6).4(C5⋊D4), (C5×C6.D4)⋊1C2, (C2×C10).50(C3⋊D4), (C2×C10).23(C2×Dic3), SmallGroup(480,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C6).D20
G = < a,b,c,d | a2=b6=c20=1, d2=a, ab=ba, cac-1=ab3, ad=da, cbc-1=dbd-1=b-1, dcd-1=ac-1 >
Subgroups: 540 in 104 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, C12, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C3×D5, C30, C30, C23⋊C4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C6.D4, C6.D4, C6×D4, C5×Dic3, C3×Dic5, Dic15, C6×D5, C2×C30, C2×C30, C23.D5, C5×C22⋊C4, C2×C5⋊D4, C23.7D6, C6×Dic5, C3×C5⋊D4, C10×Dic3, C2×Dic15, D5×C2×C6, C22×C30, C23.1D10, C5×C6.D4, C30.38D4, C6×C5⋊D4, (C2×C6).D20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C23⋊C4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C23.7D6, D5×Dic3, C15⋊D4, C3⋊D20, C23.1D10, D10⋊Dic3, (C2×C6).D20
►On 120 points
Generators in S120
(1 117)(3 119)(5 101)(7 103)(9 105)(11 107)(13 109)(15 111)(17 113)(19 115)(22 68)(24 70)(26 72)(28 74)(30 76)(32 78)(34 80)(36 62)(38 64)(40 66)(42 90)(44 92)(46 94)(48 96)(50 98)(52 100)(54 82)(56 84)(58 86)(60 88)
(1 54 40 117 82 66)(2 67 83 118 21 55)(3 56 22 119 84 68)(4 69 85 120 23 57)(5 58 24 101 86 70)(6 71 87 102 25 59)(7 60 26 103 88 72)(8 73 89 104 27 41)(9 42 28 105 90 74)(10 75 91 106 29 43)(11 44 30 107 92 76)(12 77 93 108 31 45)(13 46 32 109 94 78)(14 79 95 110 33 47)(15 48 34 111 96 80)(16 61 97 112 35 49)(17 50 36 113 98 62)(18 63 99 114 37 51)(19 52 38 115 100 64)(20 65 81 116 39 53)
(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)
(1 111 117 15)(2 110)(3 13 119 109)(4 12)(5 107 101 11)(6 106)(7 9 103 105)(10 102)(14 118)(16 20)(17 115 113 19)(18 114)(21 47)(22 94 68 46)(23 93)(24 44 70 92)(25 43)(26 90 72 42)(27 89)(28 60 74 88)(29 59)(30 86 76 58)(31 85)(32 56 78 84)(33 55)(34 82 80 54)(35 81)(36 52 62 100)(37 51)(38 98 64 50)(39 97)(40 48 66 96)(41 73)(45 69)(49 65)(53 61)(57 77)(63 99)(67 95)(71 91)(75 87)(79 83)(108 120)(112 116)
G:=sub<Sym(120)| (1,117)(3,119)(5,101)(7,103)(9,105)(11,107)(13,109)(15,111)(17,113)(19,115)(22,68)(24,70)(26,72)(28,74)(30,76)(32,78)(34,80)(36,62)(38,64)(40,66)(42,90)(44,92)(46,94)(48,96)(50,98)(52,100)(54,82)(56,84)(58,86)(60,88), (1,54,40,117,82,66)(2,67,83,118,21,55)(3,56,22,119,84,68)(4,69,85,120,23,57)(5,58,24,101,86,70)(6,71,87,102,25,59)(7,60,26,103,88,72)(8,73,89,104,27,41)(9,42,28,105,90,74)(10,75,91,106,29,43)(11,44,30,107,92,76)(12,77,93,108,31,45)(13,46,32,109,94,78)(14,79,95,110,33,47)(15,48,34,111,96,80)(16,61,97,112,35,49)(17,50,36,113,98,62)(18,63,99,114,37,51)(19,52,38,115,100,64)(20,65,81,116,39,53), (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), (1,111,117,15)(2,110)(3,13,119,109)(4,12)(5,107,101,11)(6,106)(7,9,103,105)(10,102)(14,118)(16,20)(17,115,113,19)(18,114)(21,47)(22,94,68,46)(23,93)(24,44,70,92)(25,43)(26,90,72,42)(27,89)(28,60,74,88)(29,59)(30,86,76,58)(31,85)(32,56,78,84)(33,55)(34,82,80,54)(35,81)(36,52,62,100)(37,51)(38,98,64,50)(39,97)(40,48,66,96)(41,73)(45,69)(49,65)(53,61)(57,77)(63,99)(67,95)(71,91)(75,87)(79,83)(108,120)(112,116)>;
G:=Group( (1,117)(3,119)(5,101)(7,103)(9,105)(11,107)(13,109)(15,111)(17,113)(19,115)(22,68)(24,70)(26,72)(28,74)(30,76)(32,78)(34,80)(36,62)(38,64)(40,66)(42,90)(44,92)(46,94)(48,96)(50,98)(52,100)(54,82)(56,84)(58,86)(60,88), (1,54,40,117,82,66)(2,67,83,118,21,55)(3,56,22,119,84,68)(4,69,85,120,23,57)(5,58,24,101,86,70)(6,71,87,102,25,59)(7,60,26,103,88,72)(8,73,89,104,27,41)(9,42,28,105,90,74)(10,75,91,106,29,43)(11,44,30,107,92,76)(12,77,93,108,31,45)(13,46,32,109,94,78)(14,79,95,110,33,47)(15,48,34,111,96,80)(16,61,97,112,35,49)(17,50,36,113,98,62)(18,63,99,114,37,51)(19,52,38,115,100,64)(20,65,81,116,39,53), (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), (1,111,117,15)(2,110)(3,13,119,109)(4,12)(5,107,101,11)(6,106)(7,9,103,105)(10,102)(14,118)(16,20)(17,115,113,19)(18,114)(21,47)(22,94,68,46)(23,93)(24,44,70,92)(25,43)(26,90,72,42)(27,89)(28,60,74,88)(29,59)(30,86,76,58)(31,85)(32,56,78,84)(33,55)(34,82,80,54)(35,81)(36,52,62,100)(37,51)(38,98,64,50)(39,97)(40,48,66,96)(41,73)(45,69)(49,65)(53,61)(57,77)(63,99)(67,95)(71,91)(75,87)(79,83)(108,120)(112,116) );
G=PermutationGroup([[(1,117),(3,119),(5,101),(7,103),(9,105),(11,107),(13,109),(15,111),(17,113),(19,115),(22,68),(24,70),(26,72),(28,74),(30,76),(32,78),(34,80),(36,62),(38,64),(40,66),(42,90),(44,92),(46,94),(48,96),(50,98),(52,100),(54,82),(56,84),(58,86),(60,88)], [(1,54,40,117,82,66),(2,67,83,118,21,55),(3,56,22,119,84,68),(4,69,85,120,23,57),(5,58,24,101,86,70),(6,71,87,102,25,59),(7,60,26,103,88,72),(8,73,89,104,27,41),(9,42,28,105,90,74),(10,75,91,106,29,43),(11,44,30,107,92,76),(12,77,93,108,31,45),(13,46,32,109,94,78),(14,79,95,110,33,47),(15,48,34,111,96,80),(16,61,97,112,35,49),(17,50,36,113,98,62),(18,63,99,114,37,51),(19,52,38,115,100,64),(20,65,81,116,39,53)], [(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)], [(1,111,117,15),(2,110),(3,13,119,109),(4,12),(5,107,101,11),(6,106),(7,9,103,105),(10,102),(14,118),(16,20),(17,115,113,19),(18,114),(21,47),(22,94,68,46),(23,93),(24,44,70,92),(25,43),(26,90,72,42),(27,89),(28,60,74,88),(29,59),(30,86,76,58),(31,85),(32,56,78,84),(33,55),(34,82,80,54),(35,81),(36,52,62,100),(37,51),(38,98,64,50),(39,97),(40,48,66,96),(41,73),(45,69),(49,65),(53,61),(57,77),(63,99),(67,95),(71,91),(75,87),(79,83),(108,120),(112,116)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 2 | 12 | 12 | 20 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 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 | S3 | D4 | D5 | Dic3 | Dic3 | D6 | D10 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | C23⋊C4 | S3×D5 | C23.7D6 | D5×Dic3 | C15⋊D4 | C3⋊D20 | C23.1D10 | (C2×C6).D20 |
| kernel | (C2×C6).D20 | C5×C6.D4 | C30.38D4 | C6×C5⋊D4 | C6×Dic5 | D5×C2×C6 | C2×C5⋊D4 | C2×C30 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C22×C6 | C2×C10 | C2×C6 | C2×C6 | C2×C6 | C15 | C23 | C5 | C22 | C22 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of (C2×C6).D20 ►in GL4(𝔽61) generated by
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 33 | 44 | 1 | 0 |
| 33 | 44 | 0 | 1 |
| 25 | 45 | 0 | 0 |
| 49 | 37 | 0 | 0 |
| 14 | 53 | 53 | 16 |
| 41 | 28 | 45 | 9 |
| 36 | 48 | 60 | 42 |
| 59 | 38 | 18 | 17 |
| 60 | 58 | 17 | 31 |
| 13 | 27 | 17 | 31 |
| 47 | 44 | 0 | 0 |
| 8 | 14 | 0 | 0 |
| 41 | 25 | 0 | 1 |
| 41 | 25 | 1 | 0 |
G:=sub<GL(4,GF(61))| [60,0,33,33,0,60,44,44,0,0,1,0,0,0,0,1],[25,49,14,41,45,37,53,28,0,0,53,45,0,0,16,9],[36,59,60,13,48,38,58,27,60,18,17,17,42,17,31,31],[47,8,41,41,44,14,25,25,0,0,0,1,0,0,1,0] >;
(C2×C6).D20 in GAP, Magma, Sage, TeX
(C_2\times C_6).D_{20} % in TeX
G:=Group("(C2xC6).D20"); // GroupNames label
G:=SmallGroup(480,71);
// by ID
G=gap.SmallGroup(480,71);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,422,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^6=c^20=1,d^2=a,a*b=b*a,c*a*c^-1=a*b^3,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a*c^-1>;
// generators/relations