metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊4F5, Dic30⋊2C4, D10.1D12, C15⋊1C4≀C2, C4.F5⋊1S3, (C5×D12)⋊2C4, C20.5(C4×S3), C60.3(C2×C4), C4.10(S3×F5), C12.3(C2×F5), C3⋊1(D4⋊F5), (C4×D5).23D6, (C6×D5).21D4, C2.8(D6⋊F5), C5⋊2(D12⋊C4), C10.5(D6⋊C4), C6.5(C22⋊F5), D12⋊5D5.3C2, C30.5(C22⋊C4), (C3×Dic5).24D4, (D5×C12).32C22, Dic5.26(C3⋊D4), (C4×C3⋊F5)⋊1C2, (C3×C4.F5)⋊1C2, SmallGroup(480,231)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊4F5
G = < a,b,c,d | a12=b2=c5=d4=1, bab=a-1, ac=ca, dad-1=a5, bc=cb, dbd-1=a7b, dcd-1=c3 >
Subgroups: 548 in 88 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C42, M4(2), C4○D4, Dic5, Dic5, C20, F5, D10, C2×C10, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C3×D5, C30, C4≀C2, C5⋊C8, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C2×F5, C4×Dic3, C3×M4(2), C4○D12, C3×Dic5, Dic15, C60, C3⋊F5, C6×D5, S3×C10, C4.F5, C4×F5, D4⋊2D5, D12⋊C4, C3×C5⋊C8, S3×Dic5, C15⋊D4, D5×C12, C5×D12, Dic30, C2×C3⋊F5, D4⋊F5, C3×C4.F5, C4×C3⋊F5, D12⋊5D5, D12⋊4F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, F5, C4×S3, D12, C3⋊D4, C4≀C2, C2×F5, D6⋊C4, C22⋊F5, D12⋊C4, S3×F5, D4⋊F5, D6⋊F5, D12⋊4F5
►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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 100)(14 99)(15 98)(16 97)(17 108)(18 107)(19 106)(20 105)(21 104)(22 103)(23 102)(24 101)(25 89)(26 88)(27 87)(28 86)(29 85)(30 96)(31 95)(32 94)(33 93)(34 92)(35 91)(36 90)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 82)(62 81)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 84)(72 83)
(1 85 49 23 81)(2 86 50 24 82)(3 87 51 13 83)(4 88 52 14 84)(5 89 53 15 73)(6 90 54 16 74)(7 91 55 17 75)(8 92 56 18 76)(9 93 57 19 77)(10 94 58 20 78)(11 95 59 21 79)(12 96 60 22 80)(25 110 98 70 47)(26 111 99 71 48)(27 112 100 72 37)(28 113 101 61 38)(29 114 102 62 39)(30 115 103 63 40)(31 116 104 64 41)(32 117 105 65 42)(33 118 106 66 43)(34 119 107 67 44)(35 120 108 68 45)(36 109 97 69 46)
(1 4 7 10)(2 9 8 3)(5 12 11 6)(13 86 57 76)(14 91 58 81)(15 96 59 74)(16 89 60 79)(17 94 49 84)(18 87 50 77)(19 92 51 82)(20 85 52 75)(21 90 53 80)(22 95 54 73)(23 88 55 78)(24 93 56 83)(25 114 70 102)(26 119 71 107)(27 112 72 100)(28 117 61 105)(29 110 62 98)(30 115 63 103)(31 120 64 108)(32 113 65 101)(33 118 66 106)(34 111 67 99)(35 116 68 104)(36 109 69 97)(38 42)(39 47)(41 45)(44 48)
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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,100)(14,99)(15,98)(16,97)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,89)(26,88)(27,87)(28,86)(29,85)(30,96)(31,95)(32,94)(33,93)(34,92)(35,91)(36,90)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83), (1,85,49,23,81)(2,86,50,24,82)(3,87,51,13,83)(4,88,52,14,84)(5,89,53,15,73)(6,90,54,16,74)(7,91,55,17,75)(8,92,56,18,76)(9,93,57,19,77)(10,94,58,20,78)(11,95,59,21,79)(12,96,60,22,80)(25,110,98,70,47)(26,111,99,71,48)(27,112,100,72,37)(28,113,101,61,38)(29,114,102,62,39)(30,115,103,63,40)(31,116,104,64,41)(32,117,105,65,42)(33,118,106,66,43)(34,119,107,67,44)(35,120,108,68,45)(36,109,97,69,46), (1,4,7,10)(2,9,8,3)(5,12,11,6)(13,86,57,76)(14,91,58,81)(15,96,59,74)(16,89,60,79)(17,94,49,84)(18,87,50,77)(19,92,51,82)(20,85,52,75)(21,90,53,80)(22,95,54,73)(23,88,55,78)(24,93,56,83)(25,114,70,102)(26,119,71,107)(27,112,72,100)(28,117,61,105)(29,110,62,98)(30,115,63,103)(31,120,64,108)(32,113,65,101)(33,118,66,106)(34,111,67,99)(35,116,68,104)(36,109,69,97)(38,42)(39,47)(41,45)(44,48)>;
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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,100)(14,99)(15,98)(16,97)(17,108)(18,107)(19,106)(20,105)(21,104)(22,103)(23,102)(24,101)(25,89)(26,88)(27,87)(28,86)(29,85)(30,96)(31,95)(32,94)(33,93)(34,92)(35,91)(36,90)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,120)(56,119)(57,118)(58,117)(59,116)(60,115)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,84)(72,83), (1,85,49,23,81)(2,86,50,24,82)(3,87,51,13,83)(4,88,52,14,84)(5,89,53,15,73)(6,90,54,16,74)(7,91,55,17,75)(8,92,56,18,76)(9,93,57,19,77)(10,94,58,20,78)(11,95,59,21,79)(12,96,60,22,80)(25,110,98,70,47)(26,111,99,71,48)(27,112,100,72,37)(28,113,101,61,38)(29,114,102,62,39)(30,115,103,63,40)(31,116,104,64,41)(32,117,105,65,42)(33,118,106,66,43)(34,119,107,67,44)(35,120,108,68,45)(36,109,97,69,46), (1,4,7,10)(2,9,8,3)(5,12,11,6)(13,86,57,76)(14,91,58,81)(15,96,59,74)(16,89,60,79)(17,94,49,84)(18,87,50,77)(19,92,51,82)(20,85,52,75)(21,90,53,80)(22,95,54,73)(23,88,55,78)(24,93,56,83)(25,114,70,102)(26,119,71,107)(27,112,72,100)(28,117,61,105)(29,110,62,98)(30,115,63,103)(31,120,64,108)(32,113,65,101)(33,118,66,106)(34,111,67,99)(35,116,68,104)(36,109,69,97)(38,42)(39,47)(41,45)(44,48) );
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)], [(1,39),(2,38),(3,37),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,100),(14,99),(15,98),(16,97),(17,108),(18,107),(19,106),(20,105),(21,104),(22,103),(23,102),(24,101),(25,89),(26,88),(27,87),(28,86),(29,85),(30,96),(31,95),(32,94),(33,93),(34,92),(35,91),(36,90),(49,114),(50,113),(51,112),(52,111),(53,110),(54,109),(55,120),(56,119),(57,118),(58,117),(59,116),(60,115),(61,82),(62,81),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73),(71,84),(72,83)], [(1,85,49,23,81),(2,86,50,24,82),(3,87,51,13,83),(4,88,52,14,84),(5,89,53,15,73),(6,90,54,16,74),(7,91,55,17,75),(8,92,56,18,76),(9,93,57,19,77),(10,94,58,20,78),(11,95,59,21,79),(12,96,60,22,80),(25,110,98,70,47),(26,111,99,71,48),(27,112,100,72,37),(28,113,101,61,38),(29,114,102,62,39),(30,115,103,63,40),(31,116,104,64,41),(32,117,105,65,42),(33,118,106,66,43),(34,119,107,67,44),(35,120,108,68,45),(36,109,97,69,46)], [(1,4,7,10),(2,9,8,3),(5,12,11,6),(13,86,57,76),(14,91,58,81),(15,96,59,74),(16,89,60,79),(17,94,49,84),(18,87,50,77),(19,92,51,82),(20,85,52,75),(21,90,53,80),(22,95,54,73),(23,88,55,78),(24,93,56,83),(25,114,70,102),(26,119,71,107),(27,112,72,100),(28,117,61,105),(29,110,62,98),(30,115,63,103),(31,120,64,108),(32,113,65,101),(33,118,66,106),(34,111,67,99),(35,116,68,104),(36,109,69,97),(38,42),(39,47),(41,45),(44,48)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 6A | 6B | 8A | 8B | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20 | 24A | 24B | 24C | 24D | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 24 | 24 | 24 | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 10 | 12 | 2 | 2 | 5 | 5 | 30 | 30 | 30 | 30 | 60 | 4 | 2 | 20 | 20 | 20 | 4 | 24 | 24 | 4 | 10 | 10 | 8 | 8 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | C3⋊D4 | C4×S3 | D12 | C4≀C2 | F5 | C2×F5 | C22⋊F5 | D12⋊C4 | S3×F5 | D4⋊F5 | D6⋊F5 | D12⋊4F5 |
| kernel | D12⋊4F5 | C3×C4.F5 | C4×C3⋊F5 | D12⋊5D5 | C5×D12 | Dic30 | C4.F5 | C3×Dic5 | C6×D5 | C4×D5 | Dic5 | C20 | D10 | C15 | D12 | C12 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D12⋊4F5 ►in GL8(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 205 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 212 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 31 | 29 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 99 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 142 | 142 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 177 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 71 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 177 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 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 |
G:=sub<GL(8,GF(241))| [64,205,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[212,31,0,0,0,0,0,0,4,29,0,0,0,0,0,0,0,0,99,142,0,0,0,0,0,0,43,142,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[177,71,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,64,64,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
D12⋊4F5 in GAP, Magma, Sage, TeX
D_{12}\rtimes_4F_5 % in TeX
G:=Group("D12:4F5"); // GroupNames label
G:=SmallGroup(480,231);
// by ID
G=gap.SmallGroup(480,231);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,100,675,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^5=d^4=1,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^5,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^3>;
// generators/relations