metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊3C4, C6.5D20, C2.2D60, C30.33D4, C10.5D12, C22.6D30, (C2×C60)⋊3C2, (C2×C20)⋊2S3, C5⋊3(D6⋊C4), (C2×C4)⋊1D15, (C2×C12)⋊2D5, C6.9(C4×D5), C2.5(C4×D15), C10.16(C4×S3), C15⋊7(C22⋊C4), C30.39(C2×C4), (C2×C6).24D10, (C2×C10).24D6, C3⋊2(D10⋊C4), (C2×Dic15)⋊1C2, C2.2(C15⋊7D4), C6.15(C5⋊D4), C10.15(C3⋊D4), (C2×C30).25C22, (C22×D15).1C2, SmallGroup(240,75)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊3C4
G = < a,b,c | a30=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >
Subgroups: 392 in 68 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, 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, Dic15, C60, D30, D30, C2×C30, D10⋊C4, C2×Dic15, C2×C60, C22×D15, D30⋊3C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, D15, C4×D5, D20, C5⋊D4, D6⋊C4, D30, D10⋊C4, C4×D15, D60, C15⋊7D4, D30⋊3C4
►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 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 65)(62 64)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 80)(77 79)(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)
(1 116 51 71)(2 117 52 72)(3 118 53 73)(4 119 54 74)(5 120 55 75)(6 91 56 76)(7 92 57 77)(8 93 58 78)(9 94 59 79)(10 95 60 80)(11 96 31 81)(12 97 32 82)(13 98 33 83)(14 99 34 84)(15 100 35 85)(16 101 36 86)(17 102 37 87)(18 103 38 88)(19 104 39 89)(20 105 40 90)(21 106 41 61)(22 107 42 62)(23 108 43 63)(24 109 44 64)(25 110 45 65)(26 111 46 66)(27 112 47 67)(28 113 48 68)(29 114 49 69)(30 115 50 70)
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)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(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), (1,116,51,71)(2,117,52,72)(3,118,53,73)(4,119,54,74)(5,120,55,75)(6,91,56,76)(7,92,57,77)(8,93,58,78)(9,94,59,79)(10,95,60,80)(11,96,31,81)(12,97,32,82)(13,98,33,83)(14,99,34,84)(15,100,35,85)(16,101,36,86)(17,102,37,87)(18,103,38,88)(19,104,39,89)(20,105,40,90)(21,106,41,61)(22,107,42,62)(23,108,43,63)(24,109,44,64)(25,110,45,65)(26,111,46,66)(27,112,47,67)(28,113,48,68)(29,114,49,69)(30,115,50,70)>;
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)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,65)(62,64)(66,90)(67,89)(68,88)(69,87)(70,86)(71,85)(72,84)(73,83)(74,82)(75,81)(76,80)(77,79)(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), (1,116,51,71)(2,117,52,72)(3,118,53,73)(4,119,54,74)(5,120,55,75)(6,91,56,76)(7,92,57,77)(8,93,58,78)(9,94,59,79)(10,95,60,80)(11,96,31,81)(12,97,32,82)(13,98,33,83)(14,99,34,84)(15,100,35,85)(16,101,36,86)(17,102,37,87)(18,103,38,88)(19,104,39,89)(20,105,40,90)(21,106,41,61)(22,107,42,62)(23,108,43,63)(24,109,44,64)(25,110,45,65)(26,111,46,66)(27,112,47,67)(28,113,48,68)(29,114,49,69)(30,115,50,70) );
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),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,65),(62,64),(66,90),(67,89),(68,88),(69,87),(70,86),(71,85),(72,84),(73,83),(74,82),(75,81),(76,80),(77,79),(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)], [(1,116,51,71),(2,117,52,72),(3,118,53,73),(4,119,54,74),(5,120,55,75),(6,91,56,76),(7,92,57,77),(8,93,58,78),(9,94,59,79),(10,95,60,80),(11,96,31,81),(12,97,32,82),(13,98,33,83),(14,99,34,84),(15,100,35,85),(16,101,36,86),(17,102,37,87),(18,103,38,88),(19,104,39,89),(20,105,40,90),(21,106,41,61),(22,107,42,62),(23,108,43,63),(24,109,44,64),(25,110,45,65),(26,111,46,66),(27,112,47,67),(28,113,48,68),(29,114,49,69),(30,115,50,70)]])
D30⋊3C4 is a maximal subgroup of
(C2×C20).D6 C4⋊Dic3⋊D5 C4⋊Dic5⋊S3 Dic3⋊C4⋊D5 Dic5.8D12 D6⋊Dic5⋊C2 Dic3.D20 D30.D4 C10.D4⋊S3 (C4×Dic5)⋊S3 Dic3⋊4D20 D30.C2⋊C4 D30.23(C2×C4) Dic5⋊4D12 Dic3⋊D20 D30⋊Q8 D10.16D12 Dic5⋊D12 D30⋊2Q8 D30⋊3Q8 D6.D20 D30⋊4Q8 D30.7D4 C15⋊20(C4×D4) C15⋊22(C4×D4) D10⋊D12 D6⋊D20 (C2×Dic6)⋊D5 D5×D6⋊C4 S3×D10⋊C4 D6⋊4D20 D30⋊4D4 C42⋊2D15 C4×D60 C42⋊7D15 C42⋊3D15 C22⋊C4×D15 Dic15⋊19D4 D30⋊16D4 D30.28D4 D30⋊9D4 C23.11D30 C22.D60 C4⋊C4⋊7D15 D60⋊11C4 D30.29D4 C4⋊D60 D30⋊5Q8 D30⋊6Q8 C4⋊C4⋊D15 C4×C15⋊7D4 C23.28D30 C60⋊29D4 D30⋊17D4 Dic15⋊12D4 D30⋊7Q8 C60.23D4
D30⋊3C4 is a maximal quotient of
D60⋊7C4 C23.6D30 D60⋊9C4 Dic30⋊9C4 Dic30⋊8C4 D30⋊3C8 D60⋊8C4 M4(2)⋊D15 C4.D60 D60⋊10C4 C30.29C42
66 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 | 15C | 15D | 20A | ··· | 20H | 30A | ··· | 30L | 60A | ··· | 60P |
| 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 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D10 | C4×S3 | D12 | C3⋊D4 | D15 | C4×D5 | D20 | C5⋊D4 | D30 | C4×D15 | D60 | C15⋊7D4 |
| kernel | D30⋊3C4 | C2×Dic15 | C2×C60 | C22×D15 | D30 | C2×C20 | C30 | C2×C12 | C2×C10 | C2×C6 | C10 | C10 | C10 | C2×C4 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of D30⋊3C4 ►in GL4(𝔽61) generated by
| 23 | 53 | 0 | 0 |
| 8 | 45 | 0 | 0 |
| 0 | 0 | 52 | 24 |
| 0 | 0 | 32 | 23 |
| 23 | 53 | 0 | 0 |
| 5 | 38 | 0 | 0 |
| 0 | 0 | 45 | 31 |
| 0 | 0 | 39 | 16 |
| 11 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 30 | 16 |
| 0 | 0 | 1 | 31 |
G:=sub<GL(4,GF(61))| [23,8,0,0,53,45,0,0,0,0,52,32,0,0,24,23],[23,5,0,0,53,38,0,0,0,0,45,39,0,0,31,16],[11,0,0,0,0,11,0,0,0,0,30,1,0,0,16,31] >;
D30⋊3C4 in GAP, Magma, Sage, TeX
D_{30}\rtimes_3C_4 % in TeX
G:=Group("D30:3C4"); // GroupNames label
G:=SmallGroup(240,75);
// by ID
G=gap.SmallGroup(240,75);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,964,6917]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations