metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊1F5, Dic30⋊4C4, D10.19D12, D5.2Dic12, C4.2(S3×F5), C20.4(C4×S3), C60.9(C2×C4), D5⋊C8.1S3, C5⋊(C2.Dic12), (C5×Dic6)⋊4C4, (C6×D5).20D4, C3⋊1(Q8⋊F5), (C4×D5).22D6, C2.7(D6⋊F5), C12.23(C2×F5), (C3×D5).2Q16, C60⋊C4.1C2, C10.4(D6⋊C4), C15⋊2(Q8⋊C4), (D5×Dic6).5C2, (C3×D5).4SD16, C6.4(C22⋊F5), D5.2(C24⋊C2), C30.4(C22⋊C4), (C3×Dic5).23D4, Dic5.2(C3⋊D4), (D5×C12).39C22, (C3×D5⋊C8).1C2, SmallGroup(480,230)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic30⋊C4
G = < a,b,c | a60=c4=1, b2=a30, bab-1=a-1, cac-1=a47, cbc-1=a45b >
Subgroups: 516 in 84 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C24, Dic6, Dic6, C2×Dic3, C2×C12, C3×D5, C30, Q8⋊C4, C5⋊C8, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C4⋊Dic3, C2×C24, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C3⋊F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, C2.Dic12, C3×C5⋊C8, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, C2×C3⋊F5, Q8⋊F5, C3×D5⋊C8, C60⋊C4, D5×Dic6, Dic30⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, F5, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×F5, C24⋊C2, Dic12, D6⋊C4, C22⋊F5, C2.Dic12, S3×F5, Q8⋊F5, D6⋊F5, Dic30⋊C4
►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 72 31 102)(2 71 32 101)(3 70 33 100)(4 69 34 99)(5 68 35 98)(6 67 36 97)(7 66 37 96)(8 65 38 95)(9 64 39 94)(10 63 40 93)(11 62 41 92)(12 61 42 91)(13 120 43 90)(14 119 44 89)(15 118 45 88)(16 117 46 87)(17 116 47 86)(18 115 48 85)(19 114 49 84)(20 113 50 83)(21 112 51 82)(22 111 52 81)(23 110 53 80)(24 109 54 79)(25 108 55 78)(26 107 56 77)(27 106 57 76)(28 105 58 75)(29 104 59 74)(30 103 60 73)
(2 24 50 48)(3 47 39 35)(4 10 28 22)(5 33 17 9)(6 56)(7 19 55 43)(8 42 44 30)(11 51)(12 14 60 38)(13 37 49 25)(15 23 27 59)(16 46)(18 32 54 20)(21 41)(26 36)(29 45 53 57)(34 40 58 52)(61 74 73 110)(62 97)(63 120 111 84)(64 83 100 71)(65 106 89 118)(66 69 78 105)(67 92)(68 115 116 79)(70 101 94 113)(72 87)(75 96 99 108)(76 119 88 95)(77 82)(80 91 104 103)(81 114 93 90)(85 86 109 98)(102 117)(107 112)
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), (1,72,31,102)(2,71,32,101)(3,70,33,100)(4,69,34,99)(5,68,35,98)(6,67,36,97)(7,66,37,96)(8,65,38,95)(9,64,39,94)(10,63,40,93)(11,62,41,92)(12,61,42,91)(13,120,43,90)(14,119,44,89)(15,118,45,88)(16,117,46,87)(17,116,47,86)(18,115,48,85)(19,114,49,84)(20,113,50,83)(21,112,51,82)(22,111,52,81)(23,110,53,80)(24,109,54,79)(25,108,55,78)(26,107,56,77)(27,106,57,76)(28,105,58,75)(29,104,59,74)(30,103,60,73), (2,24,50,48)(3,47,39,35)(4,10,28,22)(5,33,17,9)(6,56)(7,19,55,43)(8,42,44,30)(11,51)(12,14,60,38)(13,37,49,25)(15,23,27,59)(16,46)(18,32,54,20)(21,41)(26,36)(29,45,53,57)(34,40,58,52)(61,74,73,110)(62,97)(63,120,111,84)(64,83,100,71)(65,106,89,118)(66,69,78,105)(67,92)(68,115,116,79)(70,101,94,113)(72,87)(75,96,99,108)(76,119,88,95)(77,82)(80,91,104,103)(81,114,93,90)(85,86,109,98)(102,117)(107,112)>;
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), (1,72,31,102)(2,71,32,101)(3,70,33,100)(4,69,34,99)(5,68,35,98)(6,67,36,97)(7,66,37,96)(8,65,38,95)(9,64,39,94)(10,63,40,93)(11,62,41,92)(12,61,42,91)(13,120,43,90)(14,119,44,89)(15,118,45,88)(16,117,46,87)(17,116,47,86)(18,115,48,85)(19,114,49,84)(20,113,50,83)(21,112,51,82)(22,111,52,81)(23,110,53,80)(24,109,54,79)(25,108,55,78)(26,107,56,77)(27,106,57,76)(28,105,58,75)(29,104,59,74)(30,103,60,73), (2,24,50,48)(3,47,39,35)(4,10,28,22)(5,33,17,9)(6,56)(7,19,55,43)(8,42,44,30)(11,51)(12,14,60,38)(13,37,49,25)(15,23,27,59)(16,46)(18,32,54,20)(21,41)(26,36)(29,45,53,57)(34,40,58,52)(61,74,73,110)(62,97)(63,120,111,84)(64,83,100,71)(65,106,89,118)(66,69,78,105)(67,92)(68,115,116,79)(70,101,94,113)(72,87)(75,96,99,108)(76,119,88,95)(77,82)(80,91,104,103)(81,114,93,90)(85,86,109,98)(102,117)(107,112) );
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)], [(1,72,31,102),(2,71,32,101),(3,70,33,100),(4,69,34,99),(5,68,35,98),(6,67,36,97),(7,66,37,96),(8,65,38,95),(9,64,39,94),(10,63,40,93),(11,62,41,92),(12,61,42,91),(13,120,43,90),(14,119,44,89),(15,118,45,88),(16,117,46,87),(17,116,47,86),(18,115,48,85),(19,114,49,84),(20,113,50,83),(21,112,51,82),(22,111,52,81),(23,110,53,80),(24,109,54,79),(25,108,55,78),(26,107,56,77),(27,106,57,76),(28,105,58,75),(29,104,59,74),(30,103,60,73)], [(2,24,50,48),(3,47,39,35),(4,10,28,22),(5,33,17,9),(6,56),(7,19,55,43),(8,42,44,30),(11,51),(12,14,60,38),(13,37,49,25),(15,23,27,59),(16,46),(18,32,54,20),(21,41),(26,36),(29,45,53,57),(34,40,58,52),(61,74,73,110),(62,97),(63,120,111,84),(64,83,100,71),(65,106,89,118),(66,69,78,105),(67,92),(68,115,116,79),(70,101,94,113),(72,87),(75,96,99,108),(76,119,88,95),(77,82),(80,91,104,103),(81,114,93,90),(85,86,109,98),(102,117),(107,112)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10 | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 20C | 24A | ··· | 24H | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 10 | 12 | 60 | 60 | 60 | 4 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 2 | 2 | 10 | 10 | 8 | 8 | 24 | 24 | 10 | ··· | 10 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | - | + | - | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | SD16 | Q16 | C3⋊D4 | C4×S3 | D12 | C24⋊C2 | Dic12 | F5 | C2×F5 | C22⋊F5 | S3×F5 | Q8⋊F5 | D6⋊F5 | Dic30⋊C4 |
| kernel | Dic30⋊C4 | C3×D5⋊C8 | C60⋊C4 | D5×Dic6 | C5×Dic6 | Dic30 | D5⋊C8 | C3×Dic5 | C6×D5 | C4×D5 | C3×D5 | C3×D5 | Dic5 | C20 | D10 | D5 | D5 | Dic6 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of Dic30⋊C4 ►in GL8(𝔽241)
| 240 | 239 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 38 | 38 | 0 | 0 | 0 | 0 | 0 | 0 |
| 222 | 203 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 198 | 142 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 0 | 234 | 124 |
| 0 | 0 | 0 | 0 | 7 | 234 | 117 | 0 |
| 0 | 0 | 0 | 0 | 0 | 117 | 234 | 7 |
| 0 | 0 | 0 | 0 | 124 | 234 | 0 | 7 |
| 177 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 64 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(241))| [240,1,0,0,0,0,0,0,239,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,1,1,1,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0],[38,222,0,0,0,0,0,0,38,203,0,0,0,0,0,0,0,0,198,99,0,0,0,0,0,0,142,43,0,0,0,0,0,0,0,0,7,7,0,124,0,0,0,0,0,234,117,234,0,0,0,0,234,117,234,0,0,0,0,0,124,0,7,7],[177,64,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0] >;
Dic30⋊C4 in GAP, Magma, Sage, TeX
{\rm Dic}_{30}\rtimes C_4 % in TeX
G:=Group("Dic30:C4"); // GroupNames label
G:=SmallGroup(480,230);
// by ID
G=gap.SmallGroup(480,230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^60=c^4=1,b^2=a^30,b*a*b^-1=a^-1,c*a*c^-1=a^47,c*b*c^-1=a^45*b>;
// generators/relations