metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.6D30, C60.15D4, D60⋊23C22, C60.75C23, Dic30⋊20C22, (C6×D4)⋊2D5, (D4×C10)⋊2S3, (D4×C30)⋊2C2, (C2×D4)⋊2D15, D4⋊D15⋊13C2, (C5×D4).30D6, (C2×C4).16D30, C15⋊34(C8⋊C22), D4.D15⋊13C2, (C3×D4).30D10, (C2×C30).146D4, C30.379(C2×D4), (C2×C20).145D6, C5⋊5(D12⋊6C22), C60.7C4⋊18C2, C15⋊3C8⋊17C22, (C2×C12).144D10, C3⋊5(D4.D10), C20.40(C3⋊D4), C12.42(C5⋊D4), C4.16(C15⋊7D4), (C2×C60).71C22, D60⋊11C2⋊13C2, C4.12(C22×D15), C20.113(C22×S3), (D4×C15).35C22, C12.113(C22×D5), C22.10(C15⋊7D4), C2.9(C2×C15⋊7D4), C6.104(C2×C5⋊D4), (C2×C6).78(C5⋊D4), C10.104(C2×C3⋊D4), (C2×C10).78(C3⋊D4), SmallGroup(480,897)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.D30
G = < a,b,c,d | a4=b2=1, c30=d2=a2, bab=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=a-1b, dcd-1=c29 >
Subgroups: 692 in 136 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×C6, D15, C30, C30, C8⋊C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×D4, C5×D4, C22×C10, C4.Dic3, D4⋊S3, D4.S3, C4○D12, C6×D4, Dic15, C60, D30, C2×C30, C2×C30, C4.Dic5, D4⋊D5, D4.D5, C4○D20, D4×C10, D12⋊6C22, C15⋊3C8, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D4×C15, D4×C15, C22×C30, D4.D10, C60.7C4, D4⋊D15, D4.D15, D60⋊11C2, D4×C30, D4.D30
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C8⋊C22, C5⋊D4, C22×D5, C2×C3⋊D4, D30, C2×C5⋊D4, D12⋊6C22, C15⋊7D4, C22×D15, D4.D10, C2×C15⋊7D4, D4.D30
►On 120 points
Generators in S120
(1 16 31 46)(2 17 32 47)(3 18 33 48)(4 19 34 49)(5 20 35 50)(6 21 36 51)(7 22 37 52)(8 23 38 53)(9 24 39 54)(10 25 40 55)(11 26 41 56)(12 27 42 57)(13 28 43 58)(14 29 44 59)(15 30 45 60)(61 106 91 76)(62 107 92 77)(63 108 93 78)(64 109 94 79)(65 110 95 80)(66 111 96 81)(67 112 97 82)(68 113 98 83)(69 114 99 84)(70 115 100 85)(71 116 101 86)(72 117 102 87)(73 118 103 88)(74 119 104 89)(75 120 105 90)
(1 46)(2 17)(3 48)(4 19)(5 50)(6 21)(7 52)(8 23)(9 54)(10 25)(11 56)(12 27)(13 58)(14 29)(15 60)(16 31)(18 33)(20 35)(22 37)(24 39)(26 41)(28 43)(30 45)(32 47)(34 49)(36 51)(38 53)(40 55)(42 57)(44 59)(61 91)(63 93)(65 95)(67 97)(69 99)(71 101)(73 103)(75 105)(77 107)(79 109)(81 111)(83 113)(85 115)(87 117)(89 119)
(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 94 31 64)(2 63 32 93)(3 92 33 62)(4 61 34 91)(5 90 35 120)(6 119 36 89)(7 88 37 118)(8 117 38 87)(9 86 39 116)(10 115 40 85)(11 84 41 114)(12 113 42 83)(13 82 43 112)(14 111 44 81)(15 80 45 110)(16 109 46 79)(17 78 47 108)(18 107 48 77)(19 76 49 106)(20 105 50 75)(21 74 51 104)(22 103 52 73)(23 72 53 102)(24 101 54 71)(25 70 55 100)(26 99 56 69)(27 68 57 98)(28 97 58 67)(29 66 59 96)(30 95 60 65)
G:=sub<Sym(120)| (1,16,31,46)(2,17,32,47)(3,18,33,48)(4,19,34,49)(5,20,35,50)(6,21,36,51)(7,22,37,52)(8,23,38,53)(9,24,39,54)(10,25,40,55)(11,26,41,56)(12,27,42,57)(13,28,43,58)(14,29,44,59)(15,30,45,60)(61,106,91,76)(62,107,92,77)(63,108,93,78)(64,109,94,79)(65,110,95,80)(66,111,96,81)(67,112,97,82)(68,113,98,83)(69,114,99,84)(70,115,100,85)(71,116,101,86)(72,117,102,87)(73,118,103,88)(74,119,104,89)(75,120,105,90), (1,46)(2,17)(3,48)(4,19)(5,50)(6,21)(7,52)(8,23)(9,54)(10,25)(11,56)(12,27)(13,58)(14,29)(15,60)(16,31)(18,33)(20,35)(22,37)(24,39)(26,41)(28,43)(30,45)(32,47)(34,49)(36,51)(38,53)(40,55)(42,57)(44,59)(61,91)(63,93)(65,95)(67,97)(69,99)(71,101)(73,103)(75,105)(77,107)(79,109)(81,111)(83,113)(85,115)(87,117)(89,119), (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), (1,94,31,64)(2,63,32,93)(3,92,33,62)(4,61,34,91)(5,90,35,120)(6,119,36,89)(7,88,37,118)(8,117,38,87)(9,86,39,116)(10,115,40,85)(11,84,41,114)(12,113,42,83)(13,82,43,112)(14,111,44,81)(15,80,45,110)(16,109,46,79)(17,78,47,108)(18,107,48,77)(19,76,49,106)(20,105,50,75)(21,74,51,104)(22,103,52,73)(23,72,53,102)(24,101,54,71)(25,70,55,100)(26,99,56,69)(27,68,57,98)(28,97,58,67)(29,66,59,96)(30,95,60,65)>;
G:=Group( (1,16,31,46)(2,17,32,47)(3,18,33,48)(4,19,34,49)(5,20,35,50)(6,21,36,51)(7,22,37,52)(8,23,38,53)(9,24,39,54)(10,25,40,55)(11,26,41,56)(12,27,42,57)(13,28,43,58)(14,29,44,59)(15,30,45,60)(61,106,91,76)(62,107,92,77)(63,108,93,78)(64,109,94,79)(65,110,95,80)(66,111,96,81)(67,112,97,82)(68,113,98,83)(69,114,99,84)(70,115,100,85)(71,116,101,86)(72,117,102,87)(73,118,103,88)(74,119,104,89)(75,120,105,90), (1,46)(2,17)(3,48)(4,19)(5,50)(6,21)(7,52)(8,23)(9,54)(10,25)(11,56)(12,27)(13,58)(14,29)(15,60)(16,31)(18,33)(20,35)(22,37)(24,39)(26,41)(28,43)(30,45)(32,47)(34,49)(36,51)(38,53)(40,55)(42,57)(44,59)(61,91)(63,93)(65,95)(67,97)(69,99)(71,101)(73,103)(75,105)(77,107)(79,109)(81,111)(83,113)(85,115)(87,117)(89,119), (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), (1,94,31,64)(2,63,32,93)(3,92,33,62)(4,61,34,91)(5,90,35,120)(6,119,36,89)(7,88,37,118)(8,117,38,87)(9,86,39,116)(10,115,40,85)(11,84,41,114)(12,113,42,83)(13,82,43,112)(14,111,44,81)(15,80,45,110)(16,109,46,79)(17,78,47,108)(18,107,48,77)(19,76,49,106)(20,105,50,75)(21,74,51,104)(22,103,52,73)(23,72,53,102)(24,101,54,71)(25,70,55,100)(26,99,56,69)(27,68,57,98)(28,97,58,67)(29,66,59,96)(30,95,60,65) );
G=PermutationGroup([[(1,16,31,46),(2,17,32,47),(3,18,33,48),(4,19,34,49),(5,20,35,50),(6,21,36,51),(7,22,37,52),(8,23,38,53),(9,24,39,54),(10,25,40,55),(11,26,41,56),(12,27,42,57),(13,28,43,58),(14,29,44,59),(15,30,45,60),(61,106,91,76),(62,107,92,77),(63,108,93,78),(64,109,94,79),(65,110,95,80),(66,111,96,81),(67,112,97,82),(68,113,98,83),(69,114,99,84),(70,115,100,85),(71,116,101,86),(72,117,102,87),(73,118,103,88),(74,119,104,89),(75,120,105,90)], [(1,46),(2,17),(3,48),(4,19),(5,50),(6,21),(7,52),(8,23),(9,54),(10,25),(11,56),(12,27),(13,58),(14,29),(15,60),(16,31),(18,33),(20,35),(22,37),(24,39),(26,41),(28,43),(30,45),(32,47),(34,49),(36,51),(38,53),(40,55),(42,57),(44,59),(61,91),(63,93),(65,95),(67,97),(69,99),(71,101),(73,103),(75,105),(77,107),(79,109),(81,111),(83,113),(85,115),(87,117),(89,119)], [(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)], [(1,94,31,64),(2,63,32,93),(3,92,33,62),(4,61,34,91),(5,90,35,120),(6,119,36,89),(7,88,37,118),(8,117,38,87),(9,86,39,116),(10,115,40,85),(11,84,41,114),(12,113,42,83),(13,82,43,112),(14,111,44,81),(15,80,45,110),(16,109,46,79),(17,78,47,108),(18,107,48,77),(19,76,49,106),(20,105,50,75),(21,74,51,104),(22,103,52,73),(23,72,53,102),(24,101,54,71),(25,70,55,100),(26,99,56,69),(27,68,57,98),(28,97,58,67),(29,66,59,96),(30,95,60,65)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 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 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 4 | 4 | 60 | 2 | 2 | 2 | 60 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 60 | 60 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 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 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | D15 | C5⋊D4 | C5⋊D4 | D30 | D30 | C15⋊7D4 | C15⋊7D4 | C8⋊C22 | D12⋊6C22 | D4.D10 | D4.D30 |
| kernel | D4.D30 | C60.7C4 | D4⋊D15 | D4.D15 | D60⋊11C2 | D4×C30 | D4×C10 | C60 | C2×C30 | C6×D4 | C2×C20 | C5×D4 | C2×C12 | C3×D4 | C20 | C2×C10 | C2×D4 | C12 | C2×C6 | C2×C4 | D4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of D4.D30 ►in GL4(𝔽241) generated by
| 0 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 10 | 0 | 0 |
| 231 | 0 | 0 | 0 |
| 0 | 0 | 0 | 217 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 217 |
| 0 | 0 | 24 | 0 |
| 0 | 10 | 0 | 0 |
| 231 | 0 | 0 | 0 |
G:=sub<GL(4,GF(241))| [0,240,0,0,1,0,0,0,0,0,0,1,0,0,240,0],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,1],[0,231,0,0,10,0,0,0,0,0,0,24,0,0,217,0],[0,0,0,231,0,0,10,0,0,24,0,0,217,0,0,0] >;
D4.D30 in GAP, Magma, Sage, TeX
D_4.D_{30} % in TeX
G:=Group("D4.D30"); // GroupNames label
G:=SmallGroup(480,897);
// by ID
G=gap.SmallGroup(480,897);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,675,185,80,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=1,c^30=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^29>;
// generators/relations