metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊18D4, (C2×C6)⋊2D20, C15⋊8C22≀C2, (C2×C30)⋊11D4, C6.92(D4×D5), C5⋊2(C23⋊2D6), (C2×Dic5)⋊5D6, C10.94(S3×D4), C6.71(C2×D20), (C22×D5)⋊4D6, C3⋊4(C22⋊D20), (C2×Dic3)⋊5D10, C30.252(C2×D4), (C23×D15)⋊6C2, D30⋊4C4⋊36C2, C23.55(S3×D5), C6.D4⋊14D5, C22⋊4(C3⋊D20), (C6×Dic5)⋊9C22, (C22×C6).48D10, (C22×C10).63D6, (C2×C30).214C23, (C10×Dic3)⋊9C22, C2.44(D10⋊D6), (C22×C30).76C22, (C22×D15).114C22, (C2×C5⋊D4)⋊9S3, (C6×C5⋊D4)⋊9C2, (D5×C2×C6)⋊3C22, (C2×C3⋊D20)⋊16C2, (C2×C10)⋊8(C3⋊D4), C2.26(C2×C3⋊D20), C10.24(C2×C3⋊D4), C22.243(C2×S3×D5), (C5×C6.D4)⋊16C2, (C2×C6).226(C22×D5), (C2×C10).226(C22×S3), SmallGroup(480,648)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊18D4
G = < a,b,c,d | a30=b2=c4=d2=1, bab=a-1, cac-1=dad=a19, cbc-1=dbd=a3b, dcd=c-1 >
Subgroups: 1884 in 260 conjugacy classes, 56 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C3×D5, D15, C30, C30, C30, C22≀C2, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, D6⋊C4, C6.D4, C2×C3⋊D4, C6×D4, S3×C23, C5×Dic3, C3×Dic5, C6×D5, D30, D30, C2×C30, C2×C30, C2×C30, D10⋊C4, C5×C22⋊C4, C2×D20, C2×C5⋊D4, C23×D5, C23⋊2D6, C3⋊D20, C6×Dic5, C3×C5⋊D4, C10×Dic3, D5×C2×C6, C22×D15, C22×D15, C22×C30, C22⋊D20, D30⋊4C4, C5×C6.D4, C2×C3⋊D20, C6×C5⋊D4, C23×D15, D30⋊18D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C22≀C2, D20, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, C23⋊2D6, C3⋊D20, C2×S3×D5, C22⋊D20, C2×C3⋊D20, D10⋊D6, D30⋊18D4
►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 51)(2 50)(3 49)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 39)(14 38)(15 37)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(61 97)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 120)(69 119)(70 118)(71 117)(72 116)(73 115)(74 114)(75 113)(76 112)(77 111)(78 110)(79 109)(80 108)(81 107)(82 106)(83 105)(84 104)(85 103)(86 102)(87 101)(88 100)(89 99)(90 98)
(1 118 52 86)(2 107 53 75)(3 96 54 64)(4 115 55 83)(5 104 56 72)(6 93 57 61)(7 112 58 80)(8 101 59 69)(9 120 60 88)(10 109 31 77)(11 98 32 66)(12 117 33 85)(13 106 34 74)(14 95 35 63)(15 114 36 82)(16 103 37 71)(17 92 38 90)(18 111 39 79)(19 100 40 68)(20 119 41 87)(21 108 42 76)(22 97 43 65)(23 116 44 84)(24 105 45 73)(25 94 46 62)(26 113 47 81)(27 102 48 70)(28 91 49 89)(29 110 50 78)(30 99 51 67)
(1 86)(2 75)(3 64)(4 83)(5 72)(6 61)(7 80)(8 69)(9 88)(10 77)(11 66)(12 85)(13 74)(14 63)(15 82)(16 71)(17 90)(18 79)(19 68)(20 87)(21 76)(22 65)(23 84)(24 73)(25 62)(26 81)(27 70)(28 89)(29 78)(30 67)(31 109)(32 98)(33 117)(34 106)(35 95)(36 114)(37 103)(38 92)(39 111)(40 100)(41 119)(42 108)(43 97)(44 116)(45 105)(46 94)(47 113)(48 102)(49 91)(50 110)(51 99)(52 118)(53 107)(54 96)(55 115)(56 104)(57 93)(58 112)(59 101)(60 120)
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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98), (1,118,52,86)(2,107,53,75)(3,96,54,64)(4,115,55,83)(5,104,56,72)(6,93,57,61)(7,112,58,80)(8,101,59,69)(9,120,60,88)(10,109,31,77)(11,98,32,66)(12,117,33,85)(13,106,34,74)(14,95,35,63)(15,114,36,82)(16,103,37,71)(17,92,38,90)(18,111,39,79)(19,100,40,68)(20,119,41,87)(21,108,42,76)(22,97,43,65)(23,116,44,84)(24,105,45,73)(25,94,46,62)(26,113,47,81)(27,102,48,70)(28,91,49,89)(29,110,50,78)(30,99,51,67), (1,86)(2,75)(3,64)(4,83)(5,72)(6,61)(7,80)(8,69)(9,88)(10,77)(11,66)(12,85)(13,74)(14,63)(15,82)(16,71)(17,90)(18,79)(19,68)(20,87)(21,76)(22,65)(23,84)(24,73)(25,62)(26,81)(27,70)(28,89)(29,78)(30,67)(31,109)(32,98)(33,117)(34,106)(35,95)(36,114)(37,103)(38,92)(39,111)(40,100)(41,119)(42,108)(43,97)(44,116)(45,105)(46,94)(47,113)(48,102)(49,91)(50,110)(51,99)(52,118)(53,107)(54,96)(55,115)(56,104)(57,93)(58,112)(59,101)(60,120)>;
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), (1,51)(2,50)(3,49)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,39)(14,38)(15,37)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(61,97)(62,96)(63,95)(64,94)(65,93)(66,92)(67,91)(68,120)(69,119)(70,118)(71,117)(72,116)(73,115)(74,114)(75,113)(76,112)(77,111)(78,110)(79,109)(80,108)(81,107)(82,106)(83,105)(84,104)(85,103)(86,102)(87,101)(88,100)(89,99)(90,98), (1,118,52,86)(2,107,53,75)(3,96,54,64)(4,115,55,83)(5,104,56,72)(6,93,57,61)(7,112,58,80)(8,101,59,69)(9,120,60,88)(10,109,31,77)(11,98,32,66)(12,117,33,85)(13,106,34,74)(14,95,35,63)(15,114,36,82)(16,103,37,71)(17,92,38,90)(18,111,39,79)(19,100,40,68)(20,119,41,87)(21,108,42,76)(22,97,43,65)(23,116,44,84)(24,105,45,73)(25,94,46,62)(26,113,47,81)(27,102,48,70)(28,91,49,89)(29,110,50,78)(30,99,51,67), (1,86)(2,75)(3,64)(4,83)(5,72)(6,61)(7,80)(8,69)(9,88)(10,77)(11,66)(12,85)(13,74)(14,63)(15,82)(16,71)(17,90)(18,79)(19,68)(20,87)(21,76)(22,65)(23,84)(24,73)(25,62)(26,81)(27,70)(28,89)(29,78)(30,67)(31,109)(32,98)(33,117)(34,106)(35,95)(36,114)(37,103)(38,92)(39,111)(40,100)(41,119)(42,108)(43,97)(44,116)(45,105)(46,94)(47,113)(48,102)(49,91)(50,110)(51,99)(52,118)(53,107)(54,96)(55,115)(56,104)(57,93)(58,112)(59,101)(60,120) );
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)], [(1,51),(2,50),(3,49),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,39),(14,38),(15,37),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(61,97),(62,96),(63,95),(64,94),(65,93),(66,92),(67,91),(68,120),(69,119),(70,118),(71,117),(72,116),(73,115),(74,114),(75,113),(76,112),(77,111),(78,110),(79,109),(80,108),(81,107),(82,106),(83,105),(84,104),(85,103),(86,102),(87,101),(88,100),(89,99),(90,98)], [(1,118,52,86),(2,107,53,75),(3,96,54,64),(4,115,55,83),(5,104,56,72),(6,93,57,61),(7,112,58,80),(8,101,59,69),(9,120,60,88),(10,109,31,77),(11,98,32,66),(12,117,33,85),(13,106,34,74),(14,95,35,63),(15,114,36,82),(16,103,37,71),(17,92,38,90),(18,111,39,79),(19,100,40,68),(20,119,41,87),(21,108,42,76),(22,97,43,65),(23,116,44,84),(24,105,45,73),(25,94,46,62),(26,113,47,81),(27,102,48,70),(28,91,49,89),(29,110,50,78),(30,99,51,67)], [(1,86),(2,75),(3,64),(4,83),(5,72),(6,61),(7,80),(8,69),(9,88),(10,77),(11,66),(12,85),(13,74),(14,63),(15,82),(16,71),(17,90),(18,79),(19,68),(20,87),(21,76),(22,65),(23,84),(24,73),(25,62),(26,81),(27,70),(28,89),(29,78),(30,67),(31,109),(32,98),(33,117),(34,106),(35,95),(36,114),(37,103),(38,92),(39,111),(40,100),(41,119),(42,108),(43,97),(44,116),(45,105),(46,94),(47,113),(48,102),(49,91),(50,110),(51,99),(52,118),(53,107),(54,96),(55,115),(56,104),(57,93),(58,112),(59,101),(60,120)]])
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 | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 30 | 30 | 30 | 30 | 2 | 12 | 12 | 20 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | C3⋊D4 | D20 | S3×D4 | S3×D5 | D4×D5 | C3⋊D20 | C2×S3×D5 | D10⋊D6 |
| kernel | D30⋊18D4 | D30⋊4C4 | C5×C6.D4 | C2×C3⋊D20 | C6×C5⋊D4 | C23×D15 | C2×C5⋊D4 | D30 | C2×C30 | C6.D4 | C2×Dic5 | C22×D5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C10 | C23 | C6 | C22 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of D30⋊18D4 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 7 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 1 |
| 0 | 0 | 0 | 0 | 17 | 17 |
| 39 | 15 | 0 | 0 | 0 | 0 |
| 49 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 54 |
| 0 | 0 | 0 | 0 | 59 | 32 |
| 39 | 15 | 0 | 0 | 0 | 0 |
| 41 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 29 | 54 |
| 0 | 0 | 0 | 0 | 59 | 32 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,0,0,44,60,0,0,0,0,1,0],[1,7,0,0,0,0,0,60,0,0,0,0,0,0,60,60,0,0,0,0,0,1,0,0,0,0,0,0,44,17,0,0,0,0,1,17],[39,49,0,0,0,0,15,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,59,0,0,0,0,54,32],[39,41,0,0,0,0,15,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,59,0,0,0,0,54,32] >;
D30⋊18D4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_{18}D_4 % in TeX
G:=Group("D30:18D4"); // GroupNames label
G:=SmallGroup(480,648);
// by ID
G=gap.SmallGroup(480,648);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,219,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=d^2=1,b*a*b=a^-1,c*a*c^-1=d*a*d=a^19,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations