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, (D4×C30)⋊7C2, (C5×D4)⋊22D6, (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, C2.8(C23×D15), C6.60(C23×D5), (D4×C15)⋊24C22, (C4×D15)⋊10C22, 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
Subgroups: 1940 in 332 conjugacy classes, 119 normal (21 characteristic)
C1, C2, C2 [×9], C3, C4 [×2], C4 [×4], C22, C22 [×4], C22 [×10], C5, S3 [×4], C6, C6 [×5], C2×C4, C2×C4 [×8], D4 [×4], D4 [×14], Q8 [×2], C23 [×2], C23 [×4], D5 [×4], C10, C10 [×5], Dic3 [×4], C12 [×2], D6 [×8], C2×C6, C2×C6 [×4], C2×C6 [×2], C15, C2×D4, C2×D4 [×8], C4○D4 [×6], Dic5 [×4], C20 [×2], D10 [×8], C2×C10, C2×C10 [×4], C2×C10 [×2], Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12, C3×D4 [×4], C22×S3 [×4], C22×C6 [×2], D15 [×4], C30, C30 [×5], 2+ (1+4), Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C5×D4 [×4], C22×D5 [×4], C22×C10 [×2], C4○D12 [×2], S3×D4 [×4], D4⋊2S3 [×4], C2×C3⋊D4 [×4], C6×D4, Dic15 [×4], C60 [×2], D30 [×4], D30 [×4], C2×C30, C2×C30 [×4], C2×C30 [×2], C4○D20 [×2], D4×D5 [×4], D4⋊2D5 [×4], C2×C5⋊D4 [×4], D4×C10, D4⋊6D6, Dic30 [×2], C4×D15 [×4], D60 [×2], C2×Dic15 [×4], C15⋊7D4 [×12], C2×C60, D4×C15 [×4], C22×D15 [×4], C22×C30 [×2], D4⋊6D10, D60⋊11C2 [×2], D4×D15 [×4], D4⋊2D15 [×4], C2×C15⋊7D4 [×4], D4×C30, D4⋊6D30
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C24, D10 [×7], C22×S3 [×7], D15, 2+ (1+4), C22×D5 [×7], S3×C23, D30 [×7], C23×D5, D4⋊6D6, C22×D15 [×7], D4⋊6D10, C23×D15, D4⋊6D30
Generators and relations
G = < a,b,c,d | a4=b2=c30=d2=1, bab=cac-1=a-1, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
►On 120 points
(1 96 32 89)(2 90 33 97)(3 98 34 61)(4 62 35 99)(5 100 36 63)(6 64 37 101)(7 102 38 65)(8 66 39 103)(9 104 40 67)(10 68 41 105)(11 106 42 69)(12 70 43 107)(13 108 44 71)(14 72 45 109)(15 110 46 73)(16 74 47 111)(17 112 48 75)(18 76 49 113)(19 114 50 77)(20 78 51 115)(21 116 52 79)(22 80 53 117)(23 118 54 81)(24 82 55 119)(25 120 56 83)(26 84 57 91)(27 92 58 85)(28 86 59 93)(29 94 60 87)(30 88 31 95)
(1 89)(2 97)(3 61)(4 99)(5 63)(6 101)(7 65)(8 103)(9 67)(10 105)(11 69)(12 107)(13 71)(14 109)(15 73)(16 111)(17 75)(18 113)(19 77)(20 115)(21 79)(22 117)(23 81)(24 119)(25 83)(26 91)(27 85)(28 93)(29 87)(30 95)(31 88)(32 96)(33 90)(34 98)(35 62)(36 100)(37 64)(38 102)(39 66)(40 104)(41 68)(42 106)(43 70)(44 108)(45 72)(46 110)(47 74)(48 112)(49 76)(50 114)(51 78)(52 116)(53 80)(54 118)(55 82)(56 120)(57 84)(58 92)(59 86)(60 94)
(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 95)(2 94)(3 93)(4 92)(5 91)(6 120)(7 119)(8 118)(9 117)(10 116)(11 115)(12 114)(13 113)(14 112)(15 111)(16 110)(17 109)(18 108)(19 107)(20 106)(21 105)(22 104)(23 103)(24 102)(25 101)(26 100)(27 99)(28 98)(29 97)(30 96)(31 89)(32 88)(33 87)(34 86)(35 85)(36 84)(37 83)(38 82)(39 81)(40 80)(41 79)(42 78)(43 77)(44 76)(45 75)(46 74)(47 73)(48 72)(49 71)(50 70)(51 69)(52 68)(53 67)(54 66)(55 65)(56 64)(57 63)(58 62)(59 61)(60 90)
G:=sub<Sym(120)| (1,96,32,89)(2,90,33,97)(3,98,34,61)(4,62,35,99)(5,100,36,63)(6,64,37,101)(7,102,38,65)(8,66,39,103)(9,104,40,67)(10,68,41,105)(11,106,42,69)(12,70,43,107)(13,108,44,71)(14,72,45,109)(15,110,46,73)(16,74,47,111)(17,112,48,75)(18,76,49,113)(19,114,50,77)(20,78,51,115)(21,116,52,79)(22,80,53,117)(23,118,54,81)(24,82,55,119)(25,120,56,83)(26,84,57,91)(27,92,58,85)(28,86,59,93)(29,94,60,87)(30,88,31,95), (1,89)(2,97)(3,61)(4,99)(5,63)(6,101)(7,65)(8,103)(9,67)(10,105)(11,69)(12,107)(13,71)(14,109)(15,73)(16,111)(17,75)(18,113)(19,77)(20,115)(21,79)(22,117)(23,81)(24,119)(25,83)(26,91)(27,85)(28,93)(29,87)(30,95)(31,88)(32,96)(33,90)(34,98)(35,62)(36,100)(37,64)(38,102)(39,66)(40,104)(41,68)(42,106)(43,70)(44,108)(45,72)(46,110)(47,74)(48,112)(49,76)(50,114)(51,78)(52,116)(53,80)(54,118)(55,82)(56,120)(57,84)(58,92)(59,86)(60,94), (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,120)(7,119)(8,118)(9,117)(10,116)(11,115)(12,114)(13,113)(14,112)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,98)(29,97)(30,96)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,90)>;
G:=Group( (1,96,32,89)(2,90,33,97)(3,98,34,61)(4,62,35,99)(5,100,36,63)(6,64,37,101)(7,102,38,65)(8,66,39,103)(9,104,40,67)(10,68,41,105)(11,106,42,69)(12,70,43,107)(13,108,44,71)(14,72,45,109)(15,110,46,73)(16,74,47,111)(17,112,48,75)(18,76,49,113)(19,114,50,77)(20,78,51,115)(21,116,52,79)(22,80,53,117)(23,118,54,81)(24,82,55,119)(25,120,56,83)(26,84,57,91)(27,92,58,85)(28,86,59,93)(29,94,60,87)(30,88,31,95), (1,89)(2,97)(3,61)(4,99)(5,63)(6,101)(7,65)(8,103)(9,67)(10,105)(11,69)(12,107)(13,71)(14,109)(15,73)(16,111)(17,75)(18,113)(19,77)(20,115)(21,79)(22,117)(23,81)(24,119)(25,83)(26,91)(27,85)(28,93)(29,87)(30,95)(31,88)(32,96)(33,90)(34,98)(35,62)(36,100)(37,64)(38,102)(39,66)(40,104)(41,68)(42,106)(43,70)(44,108)(45,72)(46,110)(47,74)(48,112)(49,76)(50,114)(51,78)(52,116)(53,80)(54,118)(55,82)(56,120)(57,84)(58,92)(59,86)(60,94), (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), (1,95)(2,94)(3,93)(4,92)(5,91)(6,120)(7,119)(8,118)(9,117)(10,116)(11,115)(12,114)(13,113)(14,112)(15,111)(16,110)(17,109)(18,108)(19,107)(20,106)(21,105)(22,104)(23,103)(24,102)(25,101)(26,100)(27,99)(28,98)(29,97)(30,96)(31,89)(32,88)(33,87)(34,86)(35,85)(36,84)(37,83)(38,82)(39,81)(40,80)(41,79)(42,78)(43,77)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,70)(51,69)(52,68)(53,67)(54,66)(55,65)(56,64)(57,63)(58,62)(59,61)(60,90) );
G=PermutationGroup([(1,96,32,89),(2,90,33,97),(3,98,34,61),(4,62,35,99),(5,100,36,63),(6,64,37,101),(7,102,38,65),(8,66,39,103),(9,104,40,67),(10,68,41,105),(11,106,42,69),(12,70,43,107),(13,108,44,71),(14,72,45,109),(15,110,46,73),(16,74,47,111),(17,112,48,75),(18,76,49,113),(19,114,50,77),(20,78,51,115),(21,116,52,79),(22,80,53,117),(23,118,54,81),(24,82,55,119),(25,120,56,83),(26,84,57,91),(27,92,58,85),(28,86,59,93),(29,94,60,87),(30,88,31,95)], [(1,89),(2,97),(3,61),(4,99),(5,63),(6,101),(7,65),(8,103),(9,67),(10,105),(11,69),(12,107),(13,71),(14,109),(15,73),(16,111),(17,75),(18,113),(19,77),(20,115),(21,79),(22,117),(23,81),(24,119),(25,83),(26,91),(27,85),(28,93),(29,87),(30,95),(31,88),(32,96),(33,90),(34,98),(35,62),(36,100),(37,64),(38,102),(39,66),(40,104),(41,68),(42,106),(43,70),(44,108),(45,72),(46,110),(47,74),(48,112),(49,76),(50,114),(51,78),(52,116),(53,80),(54,118),(55,82),(56,120),(57,84),(58,92),(59,86),(60,94)], [(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)], [(1,95),(2,94),(3,93),(4,92),(5,91),(6,120),(7,119),(8,118),(9,117),(10,116),(11,115),(12,114),(13,113),(14,112),(15,111),(16,110),(17,109),(18,108),(19,107),(20,106),(21,105),(22,104),(23,103),(24,102),(25,101),(26,100),(27,99),(28,98),(29,97),(30,96),(31,89),(32,88),(33,87),(34,86),(35,85),(36,84),(37,83),(38,82),(39,81),(40,80),(41,79),(42,78),(43,77),(44,76),(45,75),(46,74),(47,73),(48,72),(49,71),(50,70),(51,69),(52,68),(53,67),(54,66),(55,65),(56,64),(57,63),(58,62),(59,61),(60,90)])
Matrix representation ►G ⊆ 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] >;
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 |
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