metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊4C4, D12⋊1F5, D5.2D24, D10.18D12, D5⋊C8⋊1S3, C5⋊(C2.D24), C4.1(S3×F5), (C5×D12)⋊4C4, C20.2(C4×S3), C60.8(C2×C4), C60⋊C4⋊1C2, (C3×D5).2D8, C3⋊1(D20⋊C4), (C6×D5).18D4, (C4×D5).20D6, (D5×D12).5C2, C2.5(D6⋊F5), C12.22(C2×F5), C15⋊2(D4⋊C4), C10.2(D6⋊C4), (C3×D5).2SD16, C6.2(C22⋊F5), D5.3(C24⋊C2), C30.2(C22⋊C4), (C3×Dic5).21D4, Dic5.1(C3⋊D4), (D5×C12).38C22, (C3×D5⋊C8)⋊1C2, SmallGroup(480,228)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊F5
G = < a,b,c,d | a12=b2=c5=d4=1, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=a7b, dcd-1=c3 >
Subgroups: 836 in 100 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, D5, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, Dic5, C20, F5, D10, D10, C2×C10, C24, D12, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C3×D5, D15, C30, D4⋊C4, C5⋊C8, C4×D5, D20, C5⋊D4, C5×D4, C2×F5, C22×D5, C4⋊Dic3, C2×C24, C2×D12, C3×Dic5, C60, C3⋊F5, S3×D5, C6×D5, S3×C10, D30, D5⋊C8, C4⋊F5, D4×D5, C2.D24, C3×C5⋊C8, C5⋊D12, D5×C12, C5×D12, D60, C2×C3⋊F5, C2×S3×D5, D20⋊C4, C3×D5⋊C8, C60⋊C4, D5×D12, D12⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, F5, C4×S3, D12, C3⋊D4, D4⋊C4, C2×F5, C24⋊C2, D24, D6⋊C4, C22⋊F5, C2.D24, S3×F5, D20⋊C4, D6⋊F5, D12⋊F5
►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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 72)(13 120)(14 119)(15 118)(16 117)(17 116)(18 115)(19 114)(20 113)(21 112)(22 111)(23 110)(24 109)(25 75)(26 74)(27 73)(28 84)(29 83)(30 82)(31 81)(32 80)(33 79)(34 78)(35 77)(36 76)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(91 98)(92 97)(93 108)(94 107)(95 106)(96 105)
(1 19 90 73 43)(2 20 91 74 44)(3 21 92 75 45)(4 22 93 76 46)(5 23 94 77 47)(6 24 95 78 48)(7 13 96 79 37)(8 14 85 80 38)(9 15 86 81 39)(10 16 87 82 40)(11 17 88 83 41)(12 18 89 84 42)(25 60 69 112 97)(26 49 70 113 98)(27 50 71 114 99)(28 51 72 115 100)(29 52 61 116 101)(30 53 62 117 102)(31 54 63 118 103)(32 55 64 119 104)(33 56 65 120 105)(34 57 66 109 106)(35 58 67 110 107)(36 59 68 111 108)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 87 37 82)(14 86 38 81)(15 85 39 80)(16 96 40 79)(17 95 41 78)(18 94 42 77)(19 93 43 76)(20 92 44 75)(21 91 45 74)(22 90 46 73)(23 89 47 84)(24 88 48 83)(25 118 97 54)(26 117 98 53)(27 116 99 52)(28 115 100 51)(29 114 101 50)(30 113 102 49)(31 112 103 60)(32 111 104 59)(33 110 105 58)(34 109 106 57)(35 120 107 56)(36 119 108 55)(61 71)(62 70)(63 69)(64 68)(65 67)
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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,72)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,75)(26,74)(27,73)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,108)(94,107)(95,106)(96,105), (1,19,90,73,43)(2,20,91,74,44)(3,21,92,75,45)(4,22,93,76,46)(5,23,94,77,47)(6,24,95,78,48)(7,13,96,79,37)(8,14,85,80,38)(9,15,86,81,39)(10,16,87,82,40)(11,17,88,83,41)(12,18,89,84,42)(25,60,69,112,97)(26,49,70,113,98)(27,50,71,114,99)(28,51,72,115,100)(29,52,61,116,101)(30,53,62,117,102)(31,54,63,118,103)(32,55,64,119,104)(33,56,65,120,105)(34,57,66,109,106)(35,58,67,110,107)(36,59,68,111,108), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,87,37,82)(14,86,38,81)(15,85,39,80)(16,96,40,79)(17,95,41,78)(18,94,42,77)(19,93,43,76)(20,92,44,75)(21,91,45,74)(22,90,46,73)(23,89,47,84)(24,88,48,83)(25,118,97,54)(26,117,98,53)(27,116,99,52)(28,115,100,51)(29,114,101,50)(30,113,102,49)(31,112,103,60)(32,111,104,59)(33,110,105,58)(34,109,106,57)(35,120,107,56)(36,119,108,55)(61,71)(62,70)(63,69)(64,68)(65,67)>;
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), (1,71)(2,70)(3,69)(4,68)(5,67)(6,66)(7,65)(8,64)(9,63)(10,62)(11,61)(12,72)(13,120)(14,119)(15,118)(16,117)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,75)(26,74)(27,73)(28,84)(29,83)(30,82)(31,81)(32,80)(33,79)(34,78)(35,77)(36,76)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,108)(94,107)(95,106)(96,105), (1,19,90,73,43)(2,20,91,74,44)(3,21,92,75,45)(4,22,93,76,46)(5,23,94,77,47)(6,24,95,78,48)(7,13,96,79,37)(8,14,85,80,38)(9,15,86,81,39)(10,16,87,82,40)(11,17,88,83,41)(12,18,89,84,42)(25,60,69,112,97)(26,49,70,113,98)(27,50,71,114,99)(28,51,72,115,100)(29,52,61,116,101)(30,53,62,117,102)(31,54,63,118,103)(32,55,64,119,104)(33,56,65,120,105)(34,57,66,109,106)(35,58,67,110,107)(36,59,68,111,108), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,87,37,82)(14,86,38,81)(15,85,39,80)(16,96,40,79)(17,95,41,78)(18,94,42,77)(19,93,43,76)(20,92,44,75)(21,91,45,74)(22,90,46,73)(23,89,47,84)(24,88,48,83)(25,118,97,54)(26,117,98,53)(27,116,99,52)(28,115,100,51)(29,114,101,50)(30,113,102,49)(31,112,103,60)(32,111,104,59)(33,110,105,58)(34,109,106,57)(35,120,107,56)(36,119,108,55)(61,71)(62,70)(63,69)(64,68)(65,67) );
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)], [(1,71),(2,70),(3,69),(4,68),(5,67),(6,66),(7,65),(8,64),(9,63),(10,62),(11,61),(12,72),(13,120),(14,119),(15,118),(16,117),(17,116),(18,115),(19,114),(20,113),(21,112),(22,111),(23,110),(24,109),(25,75),(26,74),(27,73),(28,84),(29,83),(30,82),(31,81),(32,80),(33,79),(34,78),(35,77),(36,76),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(91,98),(92,97),(93,108),(94,107),(95,106),(96,105)], [(1,19,90,73,43),(2,20,91,74,44),(3,21,92,75,45),(4,22,93,76,46),(5,23,94,77,47),(6,24,95,78,48),(7,13,96,79,37),(8,14,85,80,38),(9,15,86,81,39),(10,16,87,82,40),(11,17,88,83,41),(12,18,89,84,42),(25,60,69,112,97),(26,49,70,113,98),(27,50,71,114,99),(28,51,72,115,100),(29,52,61,116,101),(30,53,62,117,102),(31,54,63,118,103),(32,55,64,119,104),(33,56,65,120,105),(34,57,66,109,106),(35,58,67,110,107),(36,59,68,111,108)], [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9),(13,87,37,82),(14,86,38,81),(15,85,39,80),(16,96,40,79),(17,95,41,78),(18,94,42,77),(19,93,43,76),(20,92,44,75),(21,91,45,74),(22,90,46,73),(23,89,47,84),(24,88,48,83),(25,118,97,54),(26,117,98,53),(27,116,99,52),(28,115,100,51),(29,114,101,50),(30,113,102,49),(31,112,103,60),(32,111,104,59),(33,110,105,58),(34,109,106,57),(35,120,107,56),(36,119,108,55),(61,71),(62,70),(63,69),(64,68),(65,67)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20 | 24A | ··· | 24H | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 24 | ··· | 24 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 12 | 60 | 2 | 2 | 10 | 60 | 60 | 4 | 2 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 24 | 24 | 2 | 2 | 10 | 10 | 8 | 8 | 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 | D8 | SD16 | C3⋊D4 | C4×S3 | D12 | C24⋊C2 | D24 | F5 | C2×F5 | C22⋊F5 | S3×F5 | D20⋊C4 | D6⋊F5 | D12⋊F5 |
| kernel | D12⋊F5 | C3×D5⋊C8 | C60⋊C4 | D5×D12 | C5×D12 | D60 | D5⋊C8 | C3×Dic5 | C6×D5 | C4×D5 | C3×D5 | C3×D5 | Dic5 | C20 | D10 | D5 | D5 | D12 | 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 D12⋊F5 ►in GL6(𝔽241)
| 142 | 99 | 0 | 0 | 0 | 0 |
| 142 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 127 | 136 | 0 | 0 | 0 | 0 |
| 9 | 114 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 171 | 101 | 0 | 0 | 0 | 0 |
| 171 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 240 | 240 | 240 |
G:=sub<GL(6,GF(241))| [142,142,0,0,0,0,99,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[127,9,0,0,0,0,136,114,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,240,0,1,0,0,0,240,0,0,1,0,0,240,0,0,0],[171,171,0,0,0,0,101,70,0,0,0,0,0,0,1,0,0,240,0,0,0,0,1,240,0,0,0,0,0,240,0,0,0,1,0,240] >;
D12⋊F5 in GAP, Magma, Sage, TeX
D_{12}\rtimes F_5 % in TeX
G:=Group("D12:F5"); // GroupNames label
G:=SmallGroup(480,228);
// by ID
G=gap.SmallGroup(480,228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,219,100,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^5=d^4=1,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a^7*b,d*c*d^-1=c^3>;
// generators/relations