metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊5D4, D6⋊5D20, D10⋊5D12, D6⋊C4⋊8D5, (C6×D5)⋊5D4, (C2×C20)⋊2D6, C15⋊3C22≀C2, (C2×D60)⋊3C2, (S3×C10)⋊5D4, (C2×C12)⋊2D10, C6.27(D4×D5), C5⋊1(D6⋊D4), (C2×C60)⋊1C22, (C2×Dic5)⋊2D6, C6.28(C2×D20), C2.29(D5×D12), C30.70(C2×D4), C2.29(S3×D20), C10.27(S3×D4), D10⋊C4⋊8S3, C3⋊1(C22⋊D20), (C2×Dic3)⋊2D10, C10.29(C2×D12), D30⋊4C4⋊24C2, (C6×Dic5)⋊4C22, (C22×D5).60D6, (C2×C30).166C23, (C10×Dic3)⋊4C22, (C22×S3).51D10, (C22×D15)⋊5C22, C2.19(D10⋊D6), (C2×C4)⋊3(S3×D5), (C5×D6⋊C4)⋊8C2, (C22×S3×D5)⋊3C2, (C2×C3⋊D20)⋊10C2, (C2×C5⋊D12)⋊10C2, (C3×D10⋊C4)⋊8C2, (D5×C2×C6).43C22, C22.214(C2×S3×D5), (S3×C2×C10).43C22, (C2×C6).178(C22×D5), (C2×C10).178(C22×S3), SmallGroup(480,552)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊5D4
G = < a,b,c,d | a6=b2=c20=d2=1, bab=dad=a-1, ac=ca, cbc-1=a3b, dbd=ab, dcd=c-1 >
Subgroups: 1964 in 260 conjugacy classes, 54 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, D6⋊C4, D6⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, S3×C23, C5×Dic3, C3×Dic5, C60, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, D10⋊C4, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, D6⋊D4, C3⋊D20, C5⋊D12, C6×Dic5, C10×Dic3, D60, C2×C60, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C22⋊D20, D30⋊4C4, C3×D10⋊C4, C5×D6⋊C4, C2×C3⋊D20, C2×C5⋊D12, C2×D60, C22×S3×D5, D30⋊5D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C22≀C2, D20, C22×D5, C2×D12, S3×D4, S3×D5, C2×D20, D4×D5, D6⋊D4, C2×S3×D5, C22⋊D20, D5×D12, S3×D20, D10⋊D6, D30⋊5D4
►On 120 points
Generators in S120
(1 39 114 44 77 86)(2 40 115 45 78 87)(3 21 116 46 79 88)(4 22 117 47 80 89)(5 23 118 48 61 90)(6 24 119 49 62 91)(7 25 120 50 63 92)(8 26 101 51 64 93)(9 27 102 52 65 94)(10 28 103 53 66 95)(11 29 104 54 67 96)(12 30 105 55 68 97)(13 31 106 56 69 98)(14 32 107 57 70 99)(15 33 108 58 71 100)(16 34 109 59 72 81)(17 35 110 60 73 82)(18 36 111 41 74 83)(19 37 112 42 75 84)(20 38 113 43 76 85)
(1 104)(2 97)(3 106)(4 99)(5 108)(6 81)(7 110)(8 83)(9 112)(10 85)(11 114)(12 87)(13 116)(14 89)(15 118)(16 91)(17 120)(18 93)(19 102)(20 95)(21 31)(22 70)(23 33)(24 72)(25 35)(26 74)(27 37)(28 76)(29 39)(30 78)(32 80)(34 62)(36 64)(38 66)(40 68)(41 101)(42 94)(43 103)(44 96)(45 105)(46 98)(47 107)(48 100)(49 109)(50 82)(51 111)(52 84)(53 113)(54 86)(55 115)(56 88)(57 117)(58 90)(59 119)(60 92)(61 71)(63 73)(65 75)(67 77)(69 79)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 20)(12 19)(13 18)(14 17)(15 16)(21 93)(22 92)(23 91)(24 90)(25 89)(26 88)(27 87)(28 86)(29 85)(30 84)(31 83)(32 82)(33 81)(34 100)(35 99)(36 98)(37 97)(38 96)(39 95)(40 94)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 119)(62 118)(63 117)(64 116)(65 115)(66 114)(67 113)(68 112)(69 111)(70 110)(71 109)(72 108)(73 107)(74 106)(75 105)(76 104)(77 103)(78 102)(79 101)(80 120)
G:=sub<Sym(120)| (1,39,114,44,77,86)(2,40,115,45,78,87)(3,21,116,46,79,88)(4,22,117,47,80,89)(5,23,118,48,61,90)(6,24,119,49,62,91)(7,25,120,50,63,92)(8,26,101,51,64,93)(9,27,102,52,65,94)(10,28,103,53,66,95)(11,29,104,54,67,96)(12,30,105,55,68,97)(13,31,106,56,69,98)(14,32,107,57,70,99)(15,33,108,58,71,100)(16,34,109,59,72,81)(17,35,110,60,73,82)(18,36,111,41,74,83)(19,37,112,42,75,84)(20,38,113,43,76,85), (1,104)(2,97)(3,106)(4,99)(5,108)(6,81)(7,110)(8,83)(9,112)(10,85)(11,114)(12,87)(13,116)(14,89)(15,118)(16,91)(17,120)(18,93)(19,102)(20,95)(21,31)(22,70)(23,33)(24,72)(25,35)(26,74)(27,37)(28,76)(29,39)(30,78)(32,80)(34,62)(36,64)(38,66)(40,68)(41,101)(42,94)(43,103)(44,96)(45,105)(46,98)(47,107)(48,100)(49,109)(50,82)(51,111)(52,84)(53,113)(54,86)(55,115)(56,88)(57,117)(58,90)(59,119)(60,92)(61,71)(63,73)(65,75)(67,77)(69,79), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,119)(62,118)(63,117)(64,116)(65,115)(66,114)(67,113)(68,112)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,120)>;
G:=Group( (1,39,114,44,77,86)(2,40,115,45,78,87)(3,21,116,46,79,88)(4,22,117,47,80,89)(5,23,118,48,61,90)(6,24,119,49,62,91)(7,25,120,50,63,92)(8,26,101,51,64,93)(9,27,102,52,65,94)(10,28,103,53,66,95)(11,29,104,54,67,96)(12,30,105,55,68,97)(13,31,106,56,69,98)(14,32,107,57,70,99)(15,33,108,58,71,100)(16,34,109,59,72,81)(17,35,110,60,73,82)(18,36,111,41,74,83)(19,37,112,42,75,84)(20,38,113,43,76,85), (1,104)(2,97)(3,106)(4,99)(5,108)(6,81)(7,110)(8,83)(9,112)(10,85)(11,114)(12,87)(13,116)(14,89)(15,118)(16,91)(17,120)(18,93)(19,102)(20,95)(21,31)(22,70)(23,33)(24,72)(25,35)(26,74)(27,37)(28,76)(29,39)(30,78)(32,80)(34,62)(36,64)(38,66)(40,68)(41,101)(42,94)(43,103)(44,96)(45,105)(46,98)(47,107)(48,100)(49,109)(50,82)(51,111)(52,84)(53,113)(54,86)(55,115)(56,88)(57,117)(58,90)(59,119)(60,92)(61,71)(63,73)(65,75)(67,77)(69,79), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,20)(12,19)(13,18)(14,17)(15,16)(21,93)(22,92)(23,91)(24,90)(25,89)(26,88)(27,87)(28,86)(29,85)(30,84)(31,83)(32,82)(33,81)(34,100)(35,99)(36,98)(37,97)(38,96)(39,95)(40,94)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)(57,60)(58,59)(61,119)(62,118)(63,117)(64,116)(65,115)(66,114)(67,113)(68,112)(69,111)(70,110)(71,109)(72,108)(73,107)(74,106)(75,105)(76,104)(77,103)(78,102)(79,101)(80,120) );
G=PermutationGroup([[(1,39,114,44,77,86),(2,40,115,45,78,87),(3,21,116,46,79,88),(4,22,117,47,80,89),(5,23,118,48,61,90),(6,24,119,49,62,91),(7,25,120,50,63,92),(8,26,101,51,64,93),(9,27,102,52,65,94),(10,28,103,53,66,95),(11,29,104,54,67,96),(12,30,105,55,68,97),(13,31,106,56,69,98),(14,32,107,57,70,99),(15,33,108,58,71,100),(16,34,109,59,72,81),(17,35,110,60,73,82),(18,36,111,41,74,83),(19,37,112,42,75,84),(20,38,113,43,76,85)], [(1,104),(2,97),(3,106),(4,99),(5,108),(6,81),(7,110),(8,83),(9,112),(10,85),(11,114),(12,87),(13,116),(14,89),(15,118),(16,91),(17,120),(18,93),(19,102),(20,95),(21,31),(22,70),(23,33),(24,72),(25,35),(26,74),(27,37),(28,76),(29,39),(30,78),(32,80),(34,62),(36,64),(38,66),(40,68),(41,101),(42,94),(43,103),(44,96),(45,105),(46,98),(47,107),(48,100),(49,109),(50,82),(51,111),(52,84),(53,113),(54,86),(55,115),(56,88),(57,117),(58,90),(59,119),(60,92),(61,71),(63,73),(65,75),(67,77),(69,79)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,20),(12,19),(13,18),(14,17),(15,16),(21,93),(22,92),(23,91),(24,90),(25,89),(26,88),(27,87),(28,86),(29,85),(30,84),(31,83),(32,82),(33,81),(34,100),(35,99),(36,98),(37,97),(38,96),(39,95),(40,94),(41,56),(42,55),(43,54),(44,53),(45,52),(46,51),(47,50),(48,49),(57,60),(58,59),(61,119),(62,118),(63,117),(64,116),(65,115),(66,114),(67,113),(68,112),(69,111),(70,110),(71,109),(72,108),(73,107),(74,106),(75,105),(76,104),(77,103),(78,102),(79,101),(80,120)]])
60 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 | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| 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 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 30 | 30 | 60 | 2 | 4 | 12 | 20 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D12 | D20 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | D5×D12 | S3×D20 | D10⋊D6 |
| kernel | D30⋊5D4 | D30⋊4C4 | C3×D10⋊C4 | C5×D6⋊C4 | C2×C3⋊D20 | C2×C5⋊D12 | C2×D60 | C22×S3×D5 | D10⋊C4 | C6×D5 | S3×C10 | D30 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C2×Dic3 | C2×C12 | C22×S3 | D10 | D6 | C10 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D30⋊5D4 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 48 | 1 |
| 7 | 32 | 0 | 0 | 0 | 0 |
| 29 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 53 | 20 |
| 0 | 0 | 0 | 0 | 12 | 8 |
| 7 | 32 | 0 | 0 | 0 | 0 |
| 29 | 54 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 41 |
| 0 | 0 | 0 | 0 | 55 | 53 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,1,0,0,0,0,0,0,60,48,0,0,0,0,0,1],[7,29,0,0,0,0,32,2,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,53,12,0,0,0,0,20,8],[7,29,0,0,0,0,32,54,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,8,55,0,0,0,0,41,53] >;
D30⋊5D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_5D_4 % in TeX
G:=Group("D30:5D4"); // GroupNames label
G:=SmallGroup(480,552);
// by ID
G=gap.SmallGroup(480,552);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,254,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^20=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^3*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations