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 [×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
►On 120 points
(1 57 92 87)(2 88 93 58)(3 59 94 89)(4 90 95 60)(5 31 96 61)(6 62 97 32)(7 33 98 63)(8 64 99 34)(9 35 100 65)(10 66 101 36)(11 37 102 67)(12 68 103 38)(13 39 104 69)(14 70 105 40)(15 41 106 71)(16 72 107 42)(17 43 108 73)(18 74 109 44)(19 45 110 75)(20 76 111 46)(21 47 112 77)(22 78 113 48)(23 49 114 79)(24 80 115 50)(25 51 116 81)(26 82 117 52)(27 53 118 83)(28 84 119 54)(29 55 120 85)(30 86 91 56)
(1 87)(2 58)(3 89)(4 60)(5 61)(6 32)(7 63)(8 34)(9 65)(10 36)(11 67)(12 38)(13 69)(14 40)(15 71)(16 42)(17 73)(18 44)(19 75)(20 46)(21 77)(22 48)(23 79)(24 50)(25 81)(26 52)(27 83)(28 54)(29 85)(30 56)(31 96)(33 98)(35 100)(37 102)(39 104)(41 106)(43 108)(45 110)(47 112)(49 114)(51 116)(53 118)(55 120)(57 92)(59 94)(62 97)(64 99)(66 101)(68 103)(70 105)(72 107)(74 109)(76 111)(78 113)(80 115)(82 117)(84 119)(86 91)(88 93)(90 95)
(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 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 60)(28 59)(29 58)(30 57)(61 117)(62 116)(63 115)(64 114)(65 113)(66 112)(67 111)(68 110)(69 109)(70 108)(71 107)(72 106)(73 105)(74 104)(75 103)(76 102)(77 101)(78 100)(79 99)(80 98)(81 97)(82 96)(83 95)(84 94)(85 93)(86 92)(87 91)(88 120)(89 119)(90 118)
G:=sub<Sym(120)| (1,57,92,87)(2,88,93,58)(3,59,94,89)(4,90,95,60)(5,31,96,61)(6,62,97,32)(7,33,98,63)(8,64,99,34)(9,35,100,65)(10,66,101,36)(11,37,102,67)(12,68,103,38)(13,39,104,69)(14,70,105,40)(15,41,106,71)(16,72,107,42)(17,43,108,73)(18,74,109,44)(19,45,110,75)(20,76,111,46)(21,47,112,77)(22,78,113,48)(23,49,114,79)(24,80,115,50)(25,51,116,81)(26,82,117,52)(27,53,118,83)(28,84,119,54)(29,55,120,85)(30,86,91,56), (1,87)(2,58)(3,89)(4,60)(5,61)(6,32)(7,63)(8,34)(9,65)(10,36)(11,67)(12,38)(13,69)(14,40)(15,71)(16,42)(17,73)(18,44)(19,75)(20,46)(21,77)(22,48)(23,79)(24,50)(25,81)(26,52)(27,83)(28,54)(29,85)(30,56)(31,96)(33,98)(35,100)(37,102)(39,104)(41,106)(43,108)(45,110)(47,112)(49,114)(51,116)(53,118)(55,120)(57,92)(59,94)(62,97)(64,99)(66,101)(68,103)(70,105)(72,107)(74,109)(76,111)(78,113)(80,115)(82,117)(84,119)(86,91)(88,93)(90,95), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,60)(28,59)(29,58)(30,57)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,120)(89,119)(90,118)>;
G:=Group( (1,57,92,87)(2,88,93,58)(3,59,94,89)(4,90,95,60)(5,31,96,61)(6,62,97,32)(7,33,98,63)(8,64,99,34)(9,35,100,65)(10,66,101,36)(11,37,102,67)(12,68,103,38)(13,39,104,69)(14,70,105,40)(15,41,106,71)(16,72,107,42)(17,43,108,73)(18,74,109,44)(19,45,110,75)(20,76,111,46)(21,47,112,77)(22,78,113,48)(23,49,114,79)(24,80,115,50)(25,51,116,81)(26,82,117,52)(27,53,118,83)(28,84,119,54)(29,55,120,85)(30,86,91,56), (1,87)(2,58)(3,89)(4,60)(5,61)(6,32)(7,63)(8,34)(9,65)(10,36)(11,67)(12,38)(13,69)(14,40)(15,71)(16,42)(17,73)(18,44)(19,75)(20,46)(21,77)(22,48)(23,79)(24,50)(25,81)(26,52)(27,83)(28,54)(29,85)(30,56)(31,96)(33,98)(35,100)(37,102)(39,104)(41,106)(43,108)(45,110)(47,112)(49,114)(51,116)(53,118)(55,120)(57,92)(59,94)(62,97)(64,99)(66,101)(68,103)(70,105)(72,107)(74,109)(76,111)(78,113)(80,115)(82,117)(84,119)(86,91)(88,93)(90,95), (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), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,60)(28,59)(29,58)(30,57)(61,117)(62,116)(63,115)(64,114)(65,113)(66,112)(67,111)(68,110)(69,109)(70,108)(71,107)(72,106)(73,105)(74,104)(75,103)(76,102)(77,101)(78,100)(79,99)(80,98)(81,97)(82,96)(83,95)(84,94)(85,93)(86,92)(87,91)(88,120)(89,119)(90,118) );
G=PermutationGroup([(1,57,92,87),(2,88,93,58),(3,59,94,89),(4,90,95,60),(5,31,96,61),(6,62,97,32),(7,33,98,63),(8,64,99,34),(9,35,100,65),(10,66,101,36),(11,37,102,67),(12,68,103,38),(13,39,104,69),(14,70,105,40),(15,41,106,71),(16,72,107,42),(17,43,108,73),(18,74,109,44),(19,45,110,75),(20,76,111,46),(21,47,112,77),(22,78,113,48),(23,49,114,79),(24,80,115,50),(25,51,116,81),(26,82,117,52),(27,53,118,83),(28,84,119,54),(29,55,120,85),(30,86,91,56)], [(1,87),(2,58),(3,89),(4,60),(5,61),(6,32),(7,63),(8,34),(9,65),(10,36),(11,67),(12,38),(13,69),(14,40),(15,71),(16,42),(17,73),(18,44),(19,75),(20,46),(21,77),(22,48),(23,79),(24,50),(25,81),(26,52),(27,83),(28,54),(29,85),(30,56),(31,96),(33,98),(35,100),(37,102),(39,104),(41,106),(43,108),(45,110),(47,112),(49,114),(51,116),(53,118),(55,120),(57,92),(59,94),(62,97),(64,99),(66,101),(68,103),(70,105),(72,107),(74,109),(76,111),(78,113),(80,115),(82,117),(84,119),(86,91),(88,93),(90,95)], [(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)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,48),(10,47),(11,46),(12,45),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,36),(22,35),(23,34),(24,33),(25,32),(26,31),(27,60),(28,59),(29,58),(30,57),(61,117),(62,116),(63,115),(64,114),(65,113),(66,112),(67,111),(68,110),(69,109),(70,108),(71,107),(72,106),(73,105),(74,104),(75,103),(76,102),(77,101),(78,100),(79,99),(80,98),(81,97),(82,96),(83,95),(84,94),(85,93),(86,92),(87,91),(88,120),(89,119),(90,118)])
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