metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊6D30, C23⋊3D30, D60⋊25C22, C60.84C23, C30.60C24, C15⋊92+ 1+4, D30.26C23, Dic30⋊23C22, Dic15.28C23, (C6×D4)⋊7D5, (C2×C4)⋊3D30, (C5×D4)⋊22D6, (D4×C30)⋊7C2, (C2×D4)⋊7D15, (D4×C10)⋊7S3, (C2×C20)⋊12D6, (D4×D15)⋊11C2, (C3×D4)⋊22D10, (C2×C12)⋊12D10, C5⋊5(D4⋊6D6), (C2×C60)⋊8C22, (C22×C6)⋊9D10, C3⋊5(D4⋊6D10), (C22×C10)⋊12D6, (C2×C30).9C23, D4⋊2D15⋊11C2, C6.60(C23×D5), C2.8(C23×D15), (C4×D15)⋊10C22, (D4×C15)⋊24C22, C15⋊7D4⋊10C22, C10.60(S3×C23), (C22×C30)⋊5C22, D60⋊11C2⋊15C2, C4.21(C22×D15), C20.134(C22×S3), (C2×Dic15)⋊4C22, C12.132(C22×D5), (C22×D15)⋊3C22, C22.6(C22×D15), (C2×C15⋊7D4)⋊11C2, (C2×C6).16(C22×D5), (C2×C10).17(C22×S3), SmallGroup(480,1171)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4⋊6D30
G = < a,b,c,d | a4=b2=c30=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1940 in 332 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C2×D4, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, C30, C30, 2+ 1+4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C6×D4, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, C4○D20, D4×D5, D4⋊2D5, C2×C5⋊D4, D4×C10, D4⋊6D6, Dic30, C4×D15, D60, C2×Dic15, C15⋊7D4, C2×C60, D4×C15, C22×D15, C22×C30, D4⋊6D10, D60⋊11C2, D4×D15, D4⋊2D15, C2×C15⋊7D4, D4×C30, D4⋊6D30
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, D15, 2+ 1+4, C22×D5, S3×C23, D30, C23×D5, D4⋊6D6, C22×D15, D4⋊6D10, C23×D15, D4⋊6D30
►On 120 points
Generators in S120
(1 114 54 71)(2 72 55 115)(3 116 56 73)(4 74 57 117)(5 118 58 75)(6 76 59 119)(7 120 60 77)(8 78 31 91)(9 92 32 79)(10 80 33 93)(11 94 34 81)(12 82 35 95)(13 96 36 83)(14 84 37 97)(15 98 38 85)(16 86 39 99)(17 100 40 87)(18 88 41 101)(19 102 42 89)(20 90 43 103)(21 104 44 61)(22 62 45 105)(23 106 46 63)(24 64 47 107)(25 108 48 65)(26 66 49 109)(27 110 50 67)(28 68 51 111)(29 112 52 69)(30 70 53 113)
(1 71)(2 115)(3 73)(4 117)(5 75)(6 119)(7 77)(8 91)(9 79)(10 93)(11 81)(12 95)(13 83)(14 97)(15 85)(16 99)(17 87)(18 101)(19 89)(20 103)(21 61)(22 105)(23 63)(24 107)(25 65)(26 109)(27 67)(28 111)(29 69)(30 113)(31 78)(32 92)(33 80)(34 94)(35 82)(36 96)(37 84)(38 98)(39 86)(40 100)(41 88)(42 102)(43 90)(44 104)(45 62)(46 106)(47 64)(48 108)(49 66)(50 110)(51 68)(52 112)(53 70)(54 114)(55 72)(56 116)(57 74)(58 118)(59 76)(60 120)
(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)
(1 113)(2 112)(3 111)(4 110)(5 109)(6 108)(7 107)(8 106)(9 105)(10 104)(11 103)(12 102)(13 101)(14 100)(15 99)(16 98)(17 97)(18 96)(19 95)(20 94)(21 93)(22 92)(23 91)(24 120)(25 119)(26 118)(27 117)(28 116)(29 115)(30 114)(31 63)(32 62)(33 61)(34 90)(35 89)(36 88)(37 87)(38 86)(39 85)(40 84)(41 83)(42 82)(43 81)(44 80)(45 79)(46 78)(47 77)(48 76)(49 75)(50 74)(51 73)(52 72)(53 71)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)
G:=sub<Sym(120)| (1,114,54,71)(2,72,55,115)(3,116,56,73)(4,74,57,117)(5,118,58,75)(6,76,59,119)(7,120,60,77)(8,78,31,91)(9,92,32,79)(10,80,33,93)(11,94,34,81)(12,82,35,95)(13,96,36,83)(14,84,37,97)(15,98,38,85)(16,86,39,99)(17,100,40,87)(18,88,41,101)(19,102,42,89)(20,90,43,103)(21,104,44,61)(22,62,45,105)(23,106,46,63)(24,64,47,107)(25,108,48,65)(26,66,49,109)(27,110,50,67)(28,68,51,111)(29,112,52,69)(30,70,53,113), (1,71)(2,115)(3,73)(4,117)(5,75)(6,119)(7,77)(8,91)(9,79)(10,93)(11,81)(12,95)(13,83)(14,97)(15,85)(16,99)(17,87)(18,101)(19,89)(20,103)(21,61)(22,105)(23,63)(24,107)(25,65)(26,109)(27,67)(28,111)(29,69)(30,113)(31,78)(32,92)(33,80)(34,94)(35,82)(36,96)(37,84)(38,98)(39,86)(40,100)(41,88)(42,102)(43,90)(44,104)(45,62)(46,106)(47,64)(48,108)(49,66)(50,110)(51,68)(52,112)(53,70)(54,114)(55,72)(56,116)(57,74)(58,118)(59,76)(60,120), (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), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,63)(32,62)(33,61)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64)>;
G:=Group( (1,114,54,71)(2,72,55,115)(3,116,56,73)(4,74,57,117)(5,118,58,75)(6,76,59,119)(7,120,60,77)(8,78,31,91)(9,92,32,79)(10,80,33,93)(11,94,34,81)(12,82,35,95)(13,96,36,83)(14,84,37,97)(15,98,38,85)(16,86,39,99)(17,100,40,87)(18,88,41,101)(19,102,42,89)(20,90,43,103)(21,104,44,61)(22,62,45,105)(23,106,46,63)(24,64,47,107)(25,108,48,65)(26,66,49,109)(27,110,50,67)(28,68,51,111)(29,112,52,69)(30,70,53,113), (1,71)(2,115)(3,73)(4,117)(5,75)(6,119)(7,77)(8,91)(9,79)(10,93)(11,81)(12,95)(13,83)(14,97)(15,85)(16,99)(17,87)(18,101)(19,89)(20,103)(21,61)(22,105)(23,63)(24,107)(25,65)(26,109)(27,67)(28,111)(29,69)(30,113)(31,78)(32,92)(33,80)(34,94)(35,82)(36,96)(37,84)(38,98)(39,86)(40,100)(41,88)(42,102)(43,90)(44,104)(45,62)(46,106)(47,64)(48,108)(49,66)(50,110)(51,68)(52,112)(53,70)(54,114)(55,72)(56,116)(57,74)(58,118)(59,76)(60,120), (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), (1,113)(2,112)(3,111)(4,110)(5,109)(6,108)(7,107)(8,106)(9,105)(10,104)(11,103)(12,102)(13,101)(14,100)(15,99)(16,98)(17,97)(18,96)(19,95)(20,94)(21,93)(22,92)(23,91)(24,120)(25,119)(26,118)(27,117)(28,116)(29,115)(30,114)(31,63)(32,62)(33,61)(34,90)(35,89)(36,88)(37,87)(38,86)(39,85)(40,84)(41,83)(42,82)(43,81)(44,80)(45,79)(46,78)(47,77)(48,76)(49,75)(50,74)(51,73)(52,72)(53,71)(54,70)(55,69)(56,68)(57,67)(58,66)(59,65)(60,64) );
G=PermutationGroup([[(1,114,54,71),(2,72,55,115),(3,116,56,73),(4,74,57,117),(5,118,58,75),(6,76,59,119),(7,120,60,77),(8,78,31,91),(9,92,32,79),(10,80,33,93),(11,94,34,81),(12,82,35,95),(13,96,36,83),(14,84,37,97),(15,98,38,85),(16,86,39,99),(17,100,40,87),(18,88,41,101),(19,102,42,89),(20,90,43,103),(21,104,44,61),(22,62,45,105),(23,106,46,63),(24,64,47,107),(25,108,48,65),(26,66,49,109),(27,110,50,67),(28,68,51,111),(29,112,52,69),(30,70,53,113)], [(1,71),(2,115),(3,73),(4,117),(5,75),(6,119),(7,77),(8,91),(9,79),(10,93),(11,81),(12,95),(13,83),(14,97),(15,85),(16,99),(17,87),(18,101),(19,89),(20,103),(21,61),(22,105),(23,63),(24,107),(25,65),(26,109),(27,67),(28,111),(29,69),(30,113),(31,78),(32,92),(33,80),(34,94),(35,82),(36,96),(37,84),(38,98),(39,86),(40,100),(41,88),(42,102),(43,90),(44,104),(45,62),(46,106),(47,64),(48,108),(49,66),(50,110),(51,68),(52,112),(53,70),(54,114),(55,72),(56,116),(57,74),(58,118),(59,76),(60,120)], [(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)], [(1,113),(2,112),(3,111),(4,110),(5,109),(6,108),(7,107),(8,106),(9,105),(10,104),(11,103),(12,102),(13,101),(14,100),(15,99),(16,98),(17,97),(18,96),(19,95),(20,94),(21,93),(22,92),(23,91),(24,120),(25,119),(26,118),(27,117),(28,116),(29,115),(30,114),(31,63),(32,62),(33,61),(34,90),(35,89),(36,88),(37,87),(38,86),(39,85),(40,84),(41,83),(42,82),(43,81),(44,80),(45,79),(46,78),(47,77),(48,76),(49,75),(50,74),(51,73),(52,72),(53,71),(54,70),(55,69),(56,68),(57,67),(58,66),(59,65),(60,64)]])
87 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | ··· | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D15 | D30 | D30 | D30 | 2+ 1+4 | D4⋊6D6 | D4⋊6D10 | D4⋊6D30 |
| kernel | D4⋊6D30 | D60⋊11C2 | D4×D15 | D4⋊2D15 | C2×C15⋊7D4 | D4×C30 | D4×C10 | C6×D4 | C2×C20 | C5×D4 | C22×C10 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | C2×C4 | D4 | C23 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 4 | 4 | 4 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 8 | 4 | 4 | 4 | 16 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of D4⋊6D30 ►in GL4(𝔽61) generated by
| 14 | 45 | 38 | 16 |
| 16 | 47 | 45 | 5 |
| 38 | 37 | 47 | 16 |
| 24 | 57 | 45 | 14 |
| 14 | 45 | 38 | 16 |
| 16 | 47 | 45 | 5 |
| 0 | 0 | 47 | 16 |
| 0 | 0 | 45 | 14 |
| 34 | 37 | 0 | 0 |
| 24 | 53 | 0 | 0 |
| 29 | 27 | 27 | 24 |
| 34 | 0 | 37 | 8 |
| 31 | 36 | 29 | 58 |
| 14 | 30 | 2 | 32 |
| 0 | 0 | 44 | 1 |
| 0 | 0 | 17 | 17 |
G:=sub<GL(4,GF(61))| [14,16,38,24,45,47,37,57,38,45,47,45,16,5,16,14],[14,16,0,0,45,47,0,0,38,45,47,45,16,5,16,14],[34,24,29,34,37,53,27,0,0,0,27,37,0,0,24,8],[31,14,0,0,36,30,0,0,29,2,44,17,58,32,1,17] >;
D4⋊6D30 in GAP, Magma, Sage, TeX
D_4\rtimes_6D_{30} % in TeX
G:=Group("D4:6D30"); // GroupNames label
G:=SmallGroup(480,1171);
// by ID
G=gap.SmallGroup(480,1171);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,219,675,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^30=d^2=1,b*a*b=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations