metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊16D4, C22⋊3D60, C23.19D30, (C2×C20)⋊3D6, (C2×C4)⋊1D30, (C2×C6)⋊3D20, (C2×C30)⋊1D4, (C2×D60)⋊5C2, (C2×C12)⋊3D10, (C2×C10)⋊6D12, C2.7(D4×D15), C6.98(D4×D5), C2.7(C2×D60), C15⋊11C22≀C2, C5⋊2(D6⋊D4), (C2×C60)⋊2C22, C22⋊C4⋊2D15, C6.33(C2×D20), D30⋊3C4⋊4C2, C3⋊2(C22⋊D20), C10.100(S3×D4), C30.306(C2×D4), C10.34(C2×D12), (C23×D15)⋊1C2, (C22×C6).57D10, (C22×C10).72D6, (C2×C30).279C23, (C2×Dic15)⋊1C22, (C22×D15)⋊1C22, (C22×C30).13C22, C22.41(C22×D15), (C2×C15⋊7D4)⋊1C2, (C5×C22⋊C4)⋊3S3, (C3×C22⋊C4)⋊3D5, (C15×C22⋊C4)⋊5C2, (C2×C6).275(C22×D5), (C2×C10).274(C22×S3), SmallGroup(480,847)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊16D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=dad=a-1, ac=ca, cbc-1=a15b, dbd=a13b, dcd=c-1 >
Subgroups: 2084 in 260 conjugacy classes, 59 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22⋊C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, Dic15, C60, D30, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, D6⋊D4, D60, C2×Dic15, C15⋊7D4, C2×C60, C22×D15, C22×D15, C22×D15, C22×C30, C22⋊D20, D30⋊3C4, C15×C22⋊C4, C2×D60, C2×C15⋊7D4, C23×D15, D30⋊16D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, D15, C22≀C2, D20, C22×D5, C2×D12, S3×D4, D30, C2×D20, D4×D5, D6⋊D4, D60, C22×D15, C22⋊D20, C2×D60, D4×D15, D30⋊16D4
►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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)(55 60)(56 59)(57 58)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
(1 101 58 62)(2 102 59 63)(3 103 60 64)(4 104 31 65)(5 105 32 66)(6 106 33 67)(7 107 34 68)(8 108 35 69)(9 109 36 70)(10 110 37 71)(11 111 38 72)(12 112 39 73)(13 113 40 74)(14 114 41 75)(15 115 42 76)(16 116 43 77)(17 117 44 78)(18 118 45 79)(19 119 46 80)(20 120 47 81)(21 91 48 82)(22 92 49 83)(23 93 50 84)(24 94 51 85)(25 95 52 86)(26 96 53 87)(27 97 54 88)(28 98 55 89)(29 99 56 90)(30 100 57 61)
(1 62)(2 61)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 98)(32 97)(33 96)(34 95)(35 94)(36 93)(37 92)(38 91)(39 120)(40 119)(41 118)(42 117)(43 116)(44 115)(45 114)(46 113)(47 112)(48 111)(49 110)(50 109)(51 108)(52 107)(53 106)(54 105)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(55,60)(56,59)(57,58)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109), (1,101,58,62)(2,102,59,63)(3,103,60,64)(4,104,31,65)(5,105,32,66)(6,106,33,67)(7,107,34,68)(8,108,35,69)(9,109,36,70)(10,110,37,71)(11,111,38,72)(12,112,39,73)(13,113,40,74)(14,114,41,75)(15,115,42,76)(16,116,43,77)(17,117,44,78)(18,118,45,79)(19,119,46,80)(20,120,47,81)(21,91,48,82)(22,92,49,83)(23,93,50,84)(24,94,51,85)(25,95,52,86)(26,96,53,87)(27,97,54,88)(28,98,55,89)(29,99,56,90)(30,100,57,61), (1,62)(2,61)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,54)(32,53)(33,52)(34,51)(35,50)(36,49)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)(55,60)(56,59)(57,58)(61,77)(62,76)(63,75)(64,74)(65,73)(66,72)(67,71)(68,70)(78,90)(79,89)(80,88)(81,87)(82,86)(83,85)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109), (1,101,58,62)(2,102,59,63)(3,103,60,64)(4,104,31,65)(5,105,32,66)(6,106,33,67)(7,107,34,68)(8,108,35,69)(9,109,36,70)(10,110,37,71)(11,111,38,72)(12,112,39,73)(13,113,40,74)(14,114,41,75)(15,115,42,76)(16,116,43,77)(17,117,44,78)(18,118,45,79)(19,119,46,80)(20,120,47,81)(21,91,48,82)(22,92,49,83)(23,93,50,84)(24,94,51,85)(25,95,52,86)(26,96,53,87)(27,97,54,88)(28,98,55,89)(29,99,56,90)(30,100,57,61), (1,62)(2,61)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,98)(32,97)(33,96)(34,95)(35,94)(36,93)(37,92)(38,91)(39,120)(40,119)(41,118)(42,117)(43,116)(44,115)(45,114)(46,113)(47,112)(48,111)(49,110)(50,109)(51,108)(52,107)(53,106)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,54),(32,53),(33,52),(34,51),(35,50),(36,49),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43),(55,60),(56,59),(57,58),(61,77),(62,76),(63,75),(64,74),(65,73),(66,72),(67,71),(68,70),(78,90),(79,89),(80,88),(81,87),(82,86),(83,85),(91,95),(92,94),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109)], [(1,101,58,62),(2,102,59,63),(3,103,60,64),(4,104,31,65),(5,105,32,66),(6,106,33,67),(7,107,34,68),(8,108,35,69),(9,109,36,70),(10,110,37,71),(11,111,38,72),(12,112,39,73),(13,113,40,74),(14,114,41,75),(15,115,42,76),(16,116,43,77),(17,117,44,78),(18,118,45,79),(19,119,46,80),(20,120,47,81),(21,91,48,82),(22,92,49,83),(23,93,50,84),(24,94,51,85),(25,95,52,86),(26,96,53,87),(27,97,54,88),(28,98,55,89),(29,99,56,90),(30,100,57,61)], [(1,62),(2,61),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,98),(32,97),(33,96),(34,95),(35,94),(36,93),(37,92),(38,91),(39,120),(40,119),(41,118),(42,117),(43,116),(44,115),(45,114),(46,113),(47,112),(48,111),(49,110),(50,109),(51,108),(52,107),(53,106),(54,105),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 30M | ··· | 30T | 60A | ··· | 60P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 30 | 30 | 60 | 2 | 4 | 4 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D12 | D15 | D20 | D30 | D30 | D60 | S3×D4 | D4×D5 | D4×D15 |
| kernel | D30⋊16D4 | D30⋊3C4 | C15×C22⋊C4 | C2×D60 | C2×C15⋊7D4 | C23×D15 | C5×C22⋊C4 | D30 | C2×C30 | C3×C22⋊C4 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C2×C10 | C22⋊C4 | C2×C6 | C2×C4 | C23 | C22 | C10 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 | 2 | 4 | 8 |
Matrix representation of D30⋊16D4 ►in GL6(𝔽61)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 20 | 0 | 0 |
| 0 | 0 | 49 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 23 | 8 | 0 | 0 | 0 | 0 |
| 56 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 53 | 20 | 0 | 0 |
| 0 | 0 | 6 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 17 | 1 |
| 25 | 4 | 0 | 0 | 0 | 0 |
| 57 | 36 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 59 |
| 0 | 0 | 0 | 0 | 22 | 17 |
| 28 | 6 | 0 | 0 | 0 | 0 |
| 22 | 33 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 45 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 59 |
| 0 | 0 | 0 | 0 | 22 | 17 |
G:=sub<GL(6,GF(61))| [0,60,0,0,0,0,1,18,0,0,0,0,0,0,53,49,0,0,0,0,20,7,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[23,56,0,0,0,0,8,38,0,0,0,0,0,0,53,6,0,0,0,0,20,8,0,0,0,0,0,0,60,17,0,0,0,0,0,1],[25,57,0,0,0,0,4,36,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,44,22,0,0,0,0,59,17],[28,22,0,0,0,0,6,33,0,0,0,0,0,0,60,45,0,0,0,0,0,1,0,0,0,0,0,0,44,22,0,0,0,0,59,17] >;
D30⋊16D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_{16}D_4 % in TeX
G:=Group("D30:16D4"); // GroupNames label
G:=SmallGroup(480,847);
// by ID
G=gap.SmallGroup(480,847);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,58,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^15*b,d*b*d=a^13*b,d*c*d=c^-1>;
// generators/relations