metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊4C4, C30.15D4, C6.11D20, C10.11D12, C5⋊2(D6⋊C4), C6.3(C4×D5), (C2×C6).8D10, (C2×C10).8D6, C10.10(C4×S3), C15⋊6(C22⋊C4), C30.31(C2×C4), (C2×Dic3)⋊2D5, (C2×Dic5)⋊2S3, (C6×Dic5)⋊2C2, C6.5(C5⋊D4), C3⋊1(D10⋊C4), C22.7(S3×D5), (C10×Dic3)⋊2C2, C10.5(C3⋊D4), C2.2(C5⋊D12), C2.2(C3⋊D20), (C2×C30).5C22, C2.4(D30.C2), (C22×D15).2C2, SmallGroup(240,28)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊4C4
G = < a,b,c | a30=b2=c4=1, bab=a-1, cac-1=a11, cbc-1=a25b >
Subgroups: 368 in 68 conjugacy classes, 28 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C22⋊C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×C12, C22×S3, D15, C30, C2×Dic5, C2×C20, C22×D5, D6⋊C4, C5×Dic3, C3×Dic5, D30, D30, C2×C30, D10⋊C4, C6×Dic5, C10×Dic3, C22×D15, D30⋊4C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, D30.C2, C3⋊D20, C5⋊D12, D30⋊4C4
►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 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 60)(52 59)(53 58)(54 57)(55 56)(61 85)(62 84)(63 83)(64 82)(65 81)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)(86 90)(87 89)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)
(1 101 56 66)(2 112 57 77)(3 93 58 88)(4 104 59 69)(5 115 60 80)(6 96 31 61)(7 107 32 72)(8 118 33 83)(9 99 34 64)(10 110 35 75)(11 91 36 86)(12 102 37 67)(13 113 38 78)(14 94 39 89)(15 105 40 70)(16 116 41 81)(17 97 42 62)(18 108 43 73)(19 119 44 84)(20 100 45 65)(21 111 46 76)(22 92 47 87)(23 103 48 68)(24 114 49 79)(25 95 50 90)(26 106 51 71)(27 117 52 82)(28 98 53 63)(29 109 54 74)(30 120 55 85)
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)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109), (1,101,56,66)(2,112,57,77)(3,93,58,88)(4,104,59,69)(5,115,60,80)(6,96,31,61)(7,107,32,72)(8,118,33,83)(9,99,34,64)(10,110,35,75)(11,91,36,86)(12,102,37,67)(13,113,38,78)(14,94,39,89)(15,105,40,70)(16,116,41,81)(17,97,42,62)(18,108,43,73)(19,119,44,84)(20,100,45,65)(21,111,46,76)(22,92,47,87)(23,103,48,68)(24,114,49,79)(25,95,50,90)(26,106,51,71)(27,117,52,82)(28,98,53,63)(29,109,54,74)(30,120,55,85)>;
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)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,60)(52,59)(53,58)(54,57)(55,56)(61,85)(62,84)(63,83)(64,82)(65,81)(66,80)(67,79)(68,78)(69,77)(70,76)(71,75)(72,74)(86,90)(87,89)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109), (1,101,56,66)(2,112,57,77)(3,93,58,88)(4,104,59,69)(5,115,60,80)(6,96,31,61)(7,107,32,72)(8,118,33,83)(9,99,34,64)(10,110,35,75)(11,91,36,86)(12,102,37,67)(13,113,38,78)(14,94,39,89)(15,105,40,70)(16,116,41,81)(17,97,42,62)(18,108,43,73)(19,119,44,84)(20,100,45,65)(21,111,46,76)(22,92,47,87)(23,103,48,68)(24,114,49,79)(25,95,50,90)(26,106,51,71)(27,117,52,82)(28,98,53,63)(29,109,54,74)(30,120,55,85) );
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),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,60),(52,59),(53,58),(54,57),(55,56),(61,85),(62,84),(63,83),(64,82),(65,81),(66,80),(67,79),(68,78),(69,77),(70,76),(71,75),(72,74),(86,90),(87,89),(91,95),(92,94),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109)], [(1,101,56,66),(2,112,57,77),(3,93,58,88),(4,104,59,69),(5,115,60,80),(6,96,31,61),(7,107,32,72),(8,118,33,83),(9,99,34,64),(10,110,35,75),(11,91,36,86),(12,102,37,67),(13,113,38,78),(14,94,39,89),(15,105,40,70),(16,116,41,81),(17,97,42,62),(18,108,43,73),(19,119,44,84),(20,100,45,65),(21,111,46,76),(22,92,47,87),(23,103,48,68),(24,114,49,79),(25,95,50,90),(26,106,51,71),(27,117,52,82),(28,98,53,63),(29,109,54,74),(30,120,55,85)]])
D30⋊4C4 is a maximal subgroup of
(C2×C20).D6 C4⋊Dic3⋊D5 C4⋊Dic5⋊S3 (C4×D15)⋊8C4 Dic3⋊C4⋊D5 Dic5.8D12 D6⋊Dic5⋊C2 Dic3.D20 D30.34D4 D30.35D4 C60.47D4 C60.70D4 D30⋊8Q8 C10.D4⋊S3 D30⋊9Q8 (C4×D15)⋊10C4 (C4×Dic5)⋊S3 D30⋊10Q8 Dic15⋊13D4 D30.C2⋊C4 D30.23(C2×C4) Dic15⋊14D4 D30⋊Q8 D10.16D12 D60⋊17C4 D30⋊2Q8 D30⋊D4 D30⋊3Q8 D6.D20 D60⋊14C4 D30⋊4Q8 D30.6D4 C4×C3⋊D20 C4×C5⋊D12 C12⋊7D20 (C2×Dic6)⋊D5 D30⋊2D4 C60⋊6D4 D5×D6⋊C4 S3×D10⋊C4 D30⋊5D4 Dic15.19D4 C10.(C2×D12) C6.D4⋊D5 Dic15⋊3D4 C15⋊26(C4×D4) C15⋊28(C4×D4) Dic15⋊4D4 D30.45D4 D30.16D4 (C2×C10)⋊4D12 (C2×C6)⋊D20 D30⋊18D4 D30⋊19D4 D30⋊8D4
D30⋊4C4 is a maximal quotient of
D30⋊4C8 C60.29D4 C60.31D4 D60⋊12C4 D60⋊15C4 Dic30⋊12C4 Dic30⋊15C4 D60⋊13C4 D60⋊16C4 C30.24C42 C15⋊9(C23⋊C4)
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D10 | C4×S3 | D12 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | S3×D5 | D30.C2 | C3⋊D20 | C5⋊D12 |
| kernel | D30⋊4C4 | C6×Dic5 | C10×Dic3 | C22×D15 | D30 | C2×Dic5 | C30 | C2×Dic3 | C2×C10 | C2×C6 | C10 | C10 | C10 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of D30⋊4C4 ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 60 | 0 | 0 |
| 0 | 0 | 43 | 1 |
| 0 | 0 | 42 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 44 |
| 0 | 0 | 43 | 0 |
| 50 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 |
| 0 | 0 | 25 | 7 |
| 0 | 0 | 50 | 36 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,60,0,0,0,0,43,42,0,0,1,1],[0,1,0,0,1,0,0,0,0,0,0,43,0,0,44,0],[50,11,0,0,0,11,0,0,0,0,25,50,0,0,7,36] >;
D30⋊4C4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_4C_4 % in TeX
G:=Group("D30:4C4"); // GroupNames label
G:=SmallGroup(240,28);
// by ID
G=gap.SmallGroup(240,28);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,79,490,6917]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;
// generators/relations