metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.27D4, D6⋊C4⋊14D5, D6⋊10(C4×D5), (C2×C20)⋊16D6, C6.66(D4×D5), D10⋊12(C4×S3), (C2×C12)⋊16D10, C10.53(S3×D4), D6⋊Dic5⋊23C2, (C2×C60)⋊14C22, (C2×Dic5)⋊11D6, D30.32(C2×C4), C30.161(C2×D4), D10⋊C4⋊14S3, D15⋊2(C22⋊C4), (C2×Dic3)⋊11D10, C2.5(C20⋊D6), C30.69(C22×C4), (C6×Dic5)⋊2C22, (C22×D5).58D6, C2.4(D10⋊D6), D10⋊Dic3⋊23C2, (C2×C30).163C23, (C10×Dic3)⋊2C22, (C22×S3).49D10, (C2×Dic15)⋊29C22, (C22×D15).111C22, (C2×S3×D5)⋊3C4, C5⋊2(S3×C22⋊C4), C3⋊1(D5×C22⋊C4), C6.37(C2×C4×D5), C2.39(C4×S3×D5), (C6×D5)⋊8(C2×C4), (C2×C4×D15)⋊12C2, (C2×C4)⋊13(S3×D5), C15⋊5(C2×C22⋊C4), C10.70(S3×C2×C4), (C5×D6⋊C4)⋊14C2, (S3×C10)⋊15(C2×C4), (C22×S3×D5).3C2, C22.72(C2×S3×D5), (D5×C2×C6).41C22, (C2×D30.C2)⋊12C2, (S3×C2×C10).41C22, (C3×D10⋊C4)⋊14C2, (C2×C6).175(C22×D5), (C2×C10).175(C22×S3), SmallGroup(480,549)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.27D4
G = < a,b,c,d | a30=b2=c4=1, d2=a15, bab=a-1, ac=ca, dad-1=a11, bc=cb, dbd-1=a10b, dcd-1=a15c-1 >
Subgroups: 1676 in 264 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C2×C22⋊C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C23, C5×Dic3, C3×Dic5, Dic15, C60, S3×D5, C6×D5, C6×D5, S3×C10, S3×C10, D30, C2×C30, D10⋊C4, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C23×D5, S3×C22⋊C4, D30.C2, C6×Dic5, C10×Dic3, C4×D15, C2×Dic15, C2×C60, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, D5×C22⋊C4, D10⋊Dic3, D6⋊Dic5, C3×D10⋊C4, C5×D6⋊C4, C2×D30.C2, C2×C4×D15, C22×S3×D5, D30.27D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22⋊C4, C22×C4, C2×D4, D10, C4×S3, C22×S3, C2×C22⋊C4, C4×D5, C22×D5, S3×C2×C4, S3×D4, S3×D5, C2×C4×D5, D4×D5, S3×C22⋊C4, C2×S3×D5, D5×C22⋊C4, C4×S3×D5, C20⋊D6, D10⋊D6, D30.27D4
►On 120 points
(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 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(57 60)(58 59)(61 70)(62 69)(63 68)(64 67)(65 66)(71 90)(72 89)(73 88)(74 87)(75 86)(76 85)(77 84)(78 83)(79 82)(80 81)(91 100)(92 99)(93 98)(94 97)(95 96)(101 120)(102 119)(103 118)(104 117)(105 116)(106 115)(107 114)(108 113)(109 112)(110 111)
(1 66 59 111)(2 67 60 112)(3 68 31 113)(4 69 32 114)(5 70 33 115)(6 71 34 116)(7 72 35 117)(8 73 36 118)(9 74 37 119)(10 75 38 120)(11 76 39 91)(12 77 40 92)(13 78 41 93)(14 79 42 94)(15 80 43 95)(16 81 44 96)(17 82 45 97)(18 83 46 98)(19 84 47 99)(20 85 48 100)(21 86 49 101)(22 87 50 102)(23 88 51 103)(24 89 52 104)(25 90 53 105)(26 61 54 106)(27 62 55 107)(28 63 56 108)(29 64 57 109)(30 65 58 110)
(1 96 16 111)(2 107 17 92)(3 118 18 103)(4 99 19 114)(5 110 20 95)(6 91 21 106)(7 102 22 117)(8 113 23 98)(9 94 24 109)(10 105 25 120)(11 116 26 101)(12 97 27 112)(13 108 28 93)(14 119 29 104)(15 100 30 115)(31 73 46 88)(32 84 47 69)(33 65 48 80)(34 76 49 61)(35 87 50 72)(36 68 51 83)(37 79 52 64)(38 90 53 75)(39 71 54 86)(40 82 55 67)(41 63 56 78)(42 74 57 89)(43 85 58 70)(44 66 59 81)(45 77 60 62)
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)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,60)(58,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(91,100)(92,99)(93,98)(94,97)(95,96)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111), (1,66,59,111)(2,67,60,112)(3,68,31,113)(4,69,32,114)(5,70,33,115)(6,71,34,116)(7,72,35,117)(8,73,36,118)(9,74,37,119)(10,75,38,120)(11,76,39,91)(12,77,40,92)(13,78,41,93)(14,79,42,94)(15,80,43,95)(16,81,44,96)(17,82,45,97)(18,83,46,98)(19,84,47,99)(20,85,48,100)(21,86,49,101)(22,87,50,102)(23,88,51,103)(24,89,52,104)(25,90,53,105)(26,61,54,106)(27,62,55,107)(28,63,56,108)(29,64,57,109)(30,65,58,110), (1,96,16,111)(2,107,17,92)(3,118,18,103)(4,99,19,114)(5,110,20,95)(6,91,21,106)(7,102,22,117)(8,113,23,98)(9,94,24,109)(10,105,25,120)(11,116,26,101)(12,97,27,112)(13,108,28,93)(14,119,29,104)(15,100,30,115)(31,73,46,88)(32,84,47,69)(33,65,48,80)(34,76,49,61)(35,87,50,72)(36,68,51,83)(37,79,52,64)(38,90,53,75)(39,71,54,86)(40,82,55,67)(41,63,56,78)(42,74,57,89)(43,85,58,70)(44,66,59,81)(45,77,60,62)>;
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)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(57,60)(58,59)(61,70)(62,69)(63,68)(64,67)(65,66)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)(91,100)(92,99)(93,98)(94,97)(95,96)(101,120)(102,119)(103,118)(104,117)(105,116)(106,115)(107,114)(108,113)(109,112)(110,111), (1,66,59,111)(2,67,60,112)(3,68,31,113)(4,69,32,114)(5,70,33,115)(6,71,34,116)(7,72,35,117)(8,73,36,118)(9,74,37,119)(10,75,38,120)(11,76,39,91)(12,77,40,92)(13,78,41,93)(14,79,42,94)(15,80,43,95)(16,81,44,96)(17,82,45,97)(18,83,46,98)(19,84,47,99)(20,85,48,100)(21,86,49,101)(22,87,50,102)(23,88,51,103)(24,89,52,104)(25,90,53,105)(26,61,54,106)(27,62,55,107)(28,63,56,108)(29,64,57,109)(30,65,58,110), (1,96,16,111)(2,107,17,92)(3,118,18,103)(4,99,19,114)(5,110,20,95)(6,91,21,106)(7,102,22,117)(8,113,23,98)(9,94,24,109)(10,105,25,120)(11,116,26,101)(12,97,27,112)(13,108,28,93)(14,119,29,104)(15,100,30,115)(31,73,46,88)(32,84,47,69)(33,65,48,80)(34,76,49,61)(35,87,50,72)(36,68,51,83)(37,79,52,64)(38,90,53,75)(39,71,54,86)(40,82,55,67)(41,63,56,78)(42,74,57,89)(43,85,58,70)(44,66,59,81)(45,77,60,62) );
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),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(57,60),(58,59),(61,70),(62,69),(63,68),(64,67),(65,66),(71,90),(72,89),(73,88),(74,87),(75,86),(76,85),(77,84),(78,83),(79,82),(80,81),(91,100),(92,99),(93,98),(94,97),(95,96),(101,120),(102,119),(103,118),(104,117),(105,116),(106,115),(107,114),(108,113),(109,112),(110,111)], [(1,66,59,111),(2,67,60,112),(3,68,31,113),(4,69,32,114),(5,70,33,115),(6,71,34,116),(7,72,35,117),(8,73,36,118),(9,74,37,119),(10,75,38,120),(11,76,39,91),(12,77,40,92),(13,78,41,93),(14,79,42,94),(15,80,43,95),(16,81,44,96),(17,82,45,97),(18,83,46,98),(19,84,47,99),(20,85,48,100),(21,86,49,101),(22,87,50,102),(23,88,51,103),(24,89,52,104),(25,90,53,105),(26,61,54,106),(27,62,55,107),(28,63,56,108),(29,64,57,109),(30,65,58,110)], [(1,96,16,111),(2,107,17,92),(3,118,18,103),(4,99,19,114),(5,110,20,95),(6,91,21,106),(7,102,22,117),(8,113,23,98),(9,94,24,109),(10,105,25,120),(11,116,26,101),(12,97,27,112),(13,108,28,93),(14,119,29,104),(15,100,30,115),(31,73,46,88),(32,84,47,69),(33,65,48,80),(34,76,49,61),(35,87,50,72),(36,68,51,83),(37,79,52,64),(38,90,53,75),(39,71,54,86),(40,82,55,67),(41,63,56,78),(42,74,57,89),(43,85,58,70),(44,66,59,81),(45,77,60,62)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 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 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 30 | 30 | 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 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4 | S3 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4×S3 | C4×D5 | S3×D4 | S3×D5 | D4×D5 | C2×S3×D5 | C4×S3×D5 | C20⋊D6 | D10⋊D6 |
| kernel | D30.27D4 | D10⋊Dic3 | D6⋊Dic5 | C3×D10⋊C4 | C5×D6⋊C4 | C2×D30.C2 | C2×C4×D15 | C22×S3×D5 | C2×S3×D5 | D10⋊C4 | 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 | 8 | 1 | 4 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D30.27D4 ►in GL6(𝔽61)
| 60 | 15 | 0 | 0 | 0 | 0 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 19 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 15 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 10 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 20 |
| 0 | 0 | 0 | 0 | 6 | 60 |
G:=sub<GL(6,GF(61))| [60,12,0,0,0,0,15,2,0,0,0,0,0,0,60,19,0,0,0,0,60,18,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,15,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,20,60],[50,10,0,0,0,0,0,11,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,6,0,0,0,0,20,60] >;
D30.27D4 in GAP, Magma, Sage, TeX
D_{30}._{27}D_4 % in TeX
G:=Group("D30.27D4"); // GroupNames label
G:=SmallGroup(480,549);
// by ID
G=gap.SmallGroup(480,549);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,422,219,58,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=a^15,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^11,b*c=c*b,d*b*d^-1=a^10*b,d*c*d^-1=a^15*c^-1>;
// generators/relations