direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C30, C23⋊2C30, C60⋊14C22, C30.58C23, C4⋊(C2×C30), (C2×C4)⋊2C30, (C2×C20)⋊6C6, C20⋊4(C2×C6), (C2×C60)⋊14C2, (C2×C12)⋊6C10, C12⋊4(C2×C10), C22⋊2(C2×C30), (C2×C30)⋊8C22, (C22×C10)⋊3C6, (C22×C30)⋊1C2, (C22×C6)⋊1C10, C2.1(C22×C30), C6.11(C22×C10), C10.11(C22×C6), (C2×C6)⋊2(C2×C10), (C2×C10)⋊4(C2×C6), SmallGroup(240,186)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C30
G = < a,b,c | a30=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 140 in 108 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, D4, C23, C10, C10, C10, C12, C2×C6, C2×C6, C2×C6, C15, C2×D4, C20, C2×C10, C2×C10, C2×C10, C2×C12, C3×D4, C22×C6, C30, C30, C30, C2×C20, C5×D4, C22×C10, C6×D4, C60, C2×C30, C2×C30, C2×C30, D4×C10, C2×C60, D4×C15, C22×C30, D4×C30
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C2×C10, C3×D4, C22×C6, C30, C5×D4, C22×C10, C6×D4, C2×C30, D4×C10, D4×C15, C22×C30, D4×C30
►On 120 points
Generators in S120
(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 58 92 80)(2 59 93 81)(3 60 94 82)(4 31 95 83)(5 32 96 84)(6 33 97 85)(7 34 98 86)(8 35 99 87)(9 36 100 88)(10 37 101 89)(11 38 102 90)(12 39 103 61)(13 40 104 62)(14 41 105 63)(15 42 106 64)(16 43 107 65)(17 44 108 66)(18 45 109 67)(19 46 110 68)(20 47 111 69)(21 48 112 70)(22 49 113 71)(23 50 114 72)(24 51 115 73)(25 52 116 74)(26 53 117 75)(27 54 118 76)(28 55 119 77)(29 56 120 78)(30 57 91 79)
(1 80)(2 81)(3 82)(4 83)(5 84)(6 85)(7 86)(8 87)(9 88)(10 89)(11 90)(12 61)(13 62)(14 63)(15 64)(16 65)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 106)(43 107)(44 108)(45 109)(46 110)(47 111)(48 112)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 91)(58 92)(59 93)(60 94)
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), (1,58,92,80)(2,59,93,81)(3,60,94,82)(4,31,95,83)(5,32,96,84)(6,33,97,85)(7,34,98,86)(8,35,99,87)(9,36,100,88)(10,37,101,89)(11,38,102,90)(12,39,103,61)(13,40,104,62)(14,41,105,63)(15,42,106,64)(16,43,107,65)(17,44,108,66)(18,45,109,67)(19,46,110,68)(20,47,111,69)(21,48,112,70)(22,49,113,71)(23,50,114,72)(24,51,115,73)(25,52,116,74)(26,53,117,75)(27,54,118,76)(28,55,119,77)(29,56,120,78)(30,57,91,79), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,91)(58,92)(59,93)(60,94)>;
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), (1,58,92,80)(2,59,93,81)(3,60,94,82)(4,31,95,83)(5,32,96,84)(6,33,97,85)(7,34,98,86)(8,35,99,87)(9,36,100,88)(10,37,101,89)(11,38,102,90)(12,39,103,61)(13,40,104,62)(14,41,105,63)(15,42,106,64)(16,43,107,65)(17,44,108,66)(18,45,109,67)(19,46,110,68)(20,47,111,69)(21,48,112,70)(22,49,113,71)(23,50,114,72)(24,51,115,73)(25,52,116,74)(26,53,117,75)(27,54,118,76)(28,55,119,77)(29,56,120,78)(30,57,91,79), (1,80)(2,81)(3,82)(4,83)(5,84)(6,85)(7,86)(8,87)(9,88)(10,89)(11,90)(12,61)(13,62)(14,63)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,95)(32,96)(33,97)(34,98)(35,99)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,106)(43,107)(44,108)(45,109)(46,110)(47,111)(48,112)(49,113)(50,114)(51,115)(52,116)(53,117)(54,118)(55,119)(56,120)(57,91)(58,92)(59,93)(60,94) );
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)], [(1,58,92,80),(2,59,93,81),(3,60,94,82),(4,31,95,83),(5,32,96,84),(6,33,97,85),(7,34,98,86),(8,35,99,87),(9,36,100,88),(10,37,101,89),(11,38,102,90),(12,39,103,61),(13,40,104,62),(14,41,105,63),(15,42,106,64),(16,43,107,65),(17,44,108,66),(18,45,109,67),(19,46,110,68),(20,47,111,69),(21,48,112,70),(22,49,113,71),(23,50,114,72),(24,51,115,73),(25,52,116,74),(26,53,117,75),(27,54,118,76),(28,55,119,77),(29,56,120,78),(30,57,91,79)], [(1,80),(2,81),(3,82),(4,83),(5,84),(6,85),(7,86),(8,87),(9,88),(10,89),(11,90),(12,61),(13,62),(14,63),(15,64),(16,65),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,95),(32,96),(33,97),(34,98),(35,99),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,106),(43,107),(44,108),(45,109),(46,110),(47,111),(48,112),(49,113),(50,114),(51,115),(52,116),(53,117),(54,118),(55,119),(56,120),(57,91),(58,92),(59,93),(60,94)]])
D4×C30 is a maximal subgroup of
D4⋊Dic15 C60.8D4 C23.7D30 D4.D30 C23.22D30 C60.17D4 D30⋊17D4 C60⋊2D4 Dic15⋊12D4 C60⋊3D4 D4⋊6D30
150 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | 12B | 12C | 12D | 15A | ··· | 15H | 20A | ··· | 20H | 30A | ··· | 30X | 30Y | ··· | 30BD | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
150 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | D4 | C3×D4 | C5×D4 | D4×C15 |
| kernel | D4×C30 | C2×C60 | D4×C15 | C22×C30 | D4×C10 | C6×D4 | C2×C20 | C5×D4 | C22×C10 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C30 | C10 | C6 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 8 | 8 | 8 | 32 | 16 | 2 | 4 | 8 | 16 |
Matrix representation of D4×C30 ►in GL3(𝔽61) generated by
| 60 | 0 | 0 |
| 0 | 49 | 0 |
| 0 | 0 | 49 |
| 1 | 0 | 0 |
| 0 | 24 | 59 |
| 0 | 14 | 37 |
| 60 | 0 | 0 |
| 0 | 24 | 59 |
| 0 | 13 | 37 |
G:=sub<GL(3,GF(61))| [60,0,0,0,49,0,0,0,49],[1,0,0,0,24,14,0,59,37],[60,0,0,0,24,13,0,59,37] >;
D4×C30 in GAP, Magma, Sage, TeX
D_4\times C_{30} % in TeX
G:=Group("D4xC30"); // GroupNames label
G:=SmallGroup(240,186);
// by ID
G=gap.SmallGroup(240,186);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-5,-2,1465]);
// Polycyclic
G:=Group<a,b,c|a^30=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations