direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: F5×C2×C12, C30⋊3C42, C10⋊(C4×C12), D5⋊(C4×C12), C20⋊3(C2×C12), (C2×C20)⋊5C12, C60⋊10(C2×C4), (C2×C60)⋊10C4, (C4×D5)⋊5C12, C15⋊4(C2×C42), (D5×C12)⋊13C4, (C3×D5)⋊4C42, D5.(C22×C12), (C2×Dic5)⋊7C12, (C6×Dic5)⋊17C4, Dic5⋊6(C2×C12), D10.8(C2×C12), (C22×F5).3C6, C22.17(C6×F5), C6.48(C22×F5), C10.4(C22×C12), C30.86(C22×C4), D10.8(C22×C6), (C6×D5).67C23, (C6×F5).16C22, (D5×C12).138C22, C5⋊(C2×C4×C12), C2.2(C2×C6×F5), (C2×C6×F5).6C2, (C2×C4×D5).18C6, (D5×C2×C12).39C2, (C2×F5).5(C2×C6), (C2×C6).60(C2×F5), (C2×C30).59(C2×C4), (C6×D5).46(C2×C4), (C4×D5).35(C2×C6), (C2×C10).16(C2×C12), (C3×Dic5)⋊27(C2×C4), (C3×D5).4(C22×C4), (D5×C2×C6).149C22, (C22×D5).38(C2×C6), SmallGroup(480,1050)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C2×C12 |
Subgroups: 584 in 216 conjugacy classes, 124 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], C5, C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×17], C23, D5 [×4], C10, C10 [×2], C12 [×2], C12 [×10], C2×C6, C2×C6 [×6], C15, C42 [×4], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×8], D10 [×2], D10 [×4], C2×C10, C2×C12, C2×C12 [×17], C22×C6, C3×D5 [×4], C30, C30 [×2], C2×C42, C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×12], C22×D5, C4×C12 [×4], C22×C12 [×3], C3×Dic5 [×2], C60 [×2], C3×F5 [×8], C6×D5 [×2], C6×D5 [×4], C2×C30, C4×F5 [×4], C2×C4×D5, C22×F5 [×2], C2×C4×C12, D5×C12 [×4], C6×Dic5, C2×C60, C6×F5 [×12], D5×C2×C6, C2×C4×F5, C12×F5 [×4], D5×C2×C12, C2×C6×F5 [×2], F5×C2×C12
Quotients:
C1, C2 [×7], C3, C4 [×12], C22 [×7], C6 [×7], C2×C4 [×18], C23, C12 [×12], C2×C6 [×7], C42 [×4], C22×C4 [×3], F5, C2×C12 [×18], C22×C6, C2×C42, C2×F5 [×3], C4×C12 [×4], C22×C12 [×3], C3×F5, C4×F5 [×2], C22×F5, C2×C4×C12, C6×F5 [×3], C2×C4×F5, C12×F5 [×2], C2×C6×F5, F5×C2×C12
Generators and relations
G = < a,b,c,d | a2=b12=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►On 120 points
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)(49 69)(50 70)(51 71)(52 72)(53 61)(54 62)(55 63)(56 64)(57 65)(58 66)(59 67)(60 68)(97 118)(98 119)(99 120)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(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)
(1 107 36 78 57)(2 108 25 79 58)(3 97 26 80 59)(4 98 27 81 60)(5 99 28 82 49)(6 100 29 83 50)(7 101 30 84 51)(8 102 31 73 52)(9 103 32 74 53)(10 104 33 75 54)(11 105 34 76 55)(12 106 35 77 56)(13 71 95 110 45)(14 72 96 111 46)(15 61 85 112 47)(16 62 86 113 48)(17 63 87 114 37)(18 64 88 115 38)(19 65 89 116 39)(20 66 90 117 40)(21 67 91 118 41)(22 68 92 119 42)(23 69 93 120 43)(24 70 94 109 44)
(1 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 101 45 51)(14 102 46 52)(15 103 47 53)(16 104 48 54)(17 105 37 55)(18 106 38 56)(19 107 39 57)(20 108 40 58)(21 97 41 59)(22 98 42 60)(23 99 43 49)(24 100 44 50)(25 66 79 117)(26 67 80 118)(27 68 81 119)(28 69 82 120)(29 70 83 109)(30 71 84 110)(31 72 73 111)(32 61 74 112)(33 62 75 113)(34 63 76 114)(35 64 77 115)(36 65 78 116)
G:=sub<Sym(120)| (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(97,118)(98,119)(99,120)(100,109)(101,110)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117), (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), (1,107,36,78,57)(2,108,25,79,58)(3,97,26,80,59)(4,98,27,81,60)(5,99,28,82,49)(6,100,29,83,50)(7,101,30,84,51)(8,102,31,73,52)(9,103,32,74,53)(10,104,33,75,54)(11,105,34,76,55)(12,106,35,77,56)(13,71,95,110,45)(14,72,96,111,46)(15,61,85,112,47)(16,62,86,113,48)(17,63,87,114,37)(18,64,88,115,38)(19,65,89,116,39)(20,66,90,117,40)(21,67,91,118,41)(22,68,92,119,42)(23,69,93,120,43)(24,70,94,109,44), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,101,45,51)(14,102,46,52)(15,103,47,53)(16,104,48,54)(17,105,37,55)(18,106,38,56)(19,107,39,57)(20,108,40,58)(21,97,41,59)(22,98,42,60)(23,99,43,49)(24,100,44,50)(25,66,79,117)(26,67,80,118)(27,68,81,119)(28,69,82,120)(29,70,83,109)(30,71,84,110)(31,72,73,111)(32,61,74,112)(33,62,75,113)(34,63,76,114)(35,64,77,115)(36,65,78,116)>;
G:=Group( (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,84)(14,73)(15,74)(16,75)(17,76)(18,77)(19,78)(20,79)(21,80)(22,81)(23,82)(24,83)(25,40)(26,41)(27,42)(28,43)(29,44)(30,45)(31,46)(32,47)(33,48)(34,37)(35,38)(36,39)(49,69)(50,70)(51,71)(52,72)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)(59,67)(60,68)(97,118)(98,119)(99,120)(100,109)(101,110)(102,111)(103,112)(104,113)(105,114)(106,115)(107,116)(108,117), (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), (1,107,36,78,57)(2,108,25,79,58)(3,97,26,80,59)(4,98,27,81,60)(5,99,28,82,49)(6,100,29,83,50)(7,101,30,84,51)(8,102,31,73,52)(9,103,32,74,53)(10,104,33,75,54)(11,105,34,76,55)(12,106,35,77,56)(13,71,95,110,45)(14,72,96,111,46)(15,61,85,112,47)(16,62,86,113,48)(17,63,87,114,37)(18,64,88,115,38)(19,65,89,116,39)(20,66,90,117,40)(21,67,91,118,41)(22,68,92,119,42)(23,69,93,120,43)(24,70,94,109,44), (1,89)(2,90)(3,91)(4,92)(5,93)(6,94)(7,95)(8,96)(9,85)(10,86)(11,87)(12,88)(13,101,45,51)(14,102,46,52)(15,103,47,53)(16,104,48,54)(17,105,37,55)(18,106,38,56)(19,107,39,57)(20,108,40,58)(21,97,41,59)(22,98,42,60)(23,99,43,49)(24,100,44,50)(25,66,79,117)(26,67,80,118)(27,68,81,119)(28,69,82,120)(29,70,83,109)(30,71,84,110)(31,72,73,111)(32,61,74,112)(33,62,75,113)(34,63,76,114)(35,64,77,115)(36,65,78,116) );
G=PermutationGroup([(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,85),(10,86),(11,87),(12,88),(13,84),(14,73),(15,74),(16,75),(17,76),(18,77),(19,78),(20,79),(21,80),(22,81),(23,82),(24,83),(25,40),(26,41),(27,42),(28,43),(29,44),(30,45),(31,46),(32,47),(33,48),(34,37),(35,38),(36,39),(49,69),(50,70),(51,71),(52,72),(53,61),(54,62),(55,63),(56,64),(57,65),(58,66),(59,67),(60,68),(97,118),(98,119),(99,120),(100,109),(101,110),(102,111),(103,112),(104,113),(105,114),(106,115),(107,116),(108,117)], [(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)], [(1,107,36,78,57),(2,108,25,79,58),(3,97,26,80,59),(4,98,27,81,60),(5,99,28,82,49),(6,100,29,83,50),(7,101,30,84,51),(8,102,31,73,52),(9,103,32,74,53),(10,104,33,75,54),(11,105,34,76,55),(12,106,35,77,56),(13,71,95,110,45),(14,72,96,111,46),(15,61,85,112,47),(16,62,86,113,48),(17,63,87,114,37),(18,64,88,115,38),(19,65,89,116,39),(20,66,90,117,40),(21,67,91,118,41),(22,68,92,119,42),(23,69,93,120,43),(24,70,94,109,44)], [(1,89),(2,90),(3,91),(4,92),(5,93),(6,94),(7,95),(8,96),(9,85),(10,86),(11,87),(12,88),(13,101,45,51),(14,102,46,52),(15,103,47,53),(16,104,48,54),(17,105,37,55),(18,106,38,56),(19,107,39,57),(20,108,40,58),(21,97,41,59),(22,98,42,60),(23,99,43,49),(24,100,44,50),(25,66,79,117),(26,67,80,118),(27,68,81,119),(28,69,82,120),(29,70,83,109),(30,71,84,110),(31,72,73,111),(32,61,74,112),(33,62,75,113),(34,63,76,114),(35,64,77,115),(36,65,78,116)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 21 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 60 | 60 | 60 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[11,0,0,0,0,0,21,0,0,0,0,0,21,0,0,0,0,0,21,0,0,0,0,0,21],[1,0,0,0,0,0,60,1,0,0,0,60,0,1,0,0,60,0,0,1,0,60,0,0,0],[50,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;
120 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4X | 5 | 6A | ··· | 6F | 6G | ··· | 6N | 10A | 10B | 10C | 12A | ··· | 12H | 12I | ··· | 12AV | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | 10 | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | C12 | C12 | F5 | C2×F5 | C2×F5 | C3×F5 | C4×F5 | C6×F5 | C6×F5 | C12×F5 |
| kernel | F5×C2×C12 | C12×F5 | D5×C2×C12 | C2×C6×F5 | C2×C4×F5 | D5×C12 | C6×Dic5 | C2×C60 | C6×F5 | C4×F5 | C2×C4×D5 | C22×F5 | C4×D5 | C2×Dic5 | C2×C20 | C2×F5 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 2 | 2 | 4 | 2 | 2 | 16 | 8 | 2 | 4 | 8 | 4 | 4 | 32 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
F_5\times C_2\times C_{12} % in TeX
G:=Group("F5xC2xC12"); // GroupNames label
G:=SmallGroup(480,1050);
// by ID
G=gap.SmallGroup(480,1050);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,168,268,9414,818]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations