direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: F5×C2×C12, C30⋊3C42, C10⋊(C4×C12), D5⋊(C4×C12), (C2×C60)⋊10C4, C60⋊10(C2×C4), (C2×C20)⋊5C12, C20⋊3(C2×C12), (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), C30.86(C22×C4), C10.4(C22×C12), (C6×D5).67C23, D10.8(C22×C6), (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), (D5×C2×C6).149C22, (C3×D5).4(C22×C4), (C22×D5).38(C2×C6), SmallGroup(480,1050)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — F5×C2×C12 |
Generators and relations for F5×C2×C12
G = < a,b,c,d | a2=b12=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 584 in 216 conjugacy classes, 124 normal (28 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C2×C42, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C4×C12, C22×C12, C3×Dic5, C60, C3×F5, C6×D5, C6×D5, C2×C30, C4×F5, C2×C4×D5, C22×F5, C2×C4×C12, D5×C12, C6×Dic5, C2×C60, C6×F5, D5×C2×C6, C2×C4×F5, C12×F5, D5×C2×C12, C2×C6×F5, F5×C2×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C42, C22×C4, F5, C2×C12, C22×C6, C2×C42, C2×F5, C4×C12, C22×C12, C3×F5, C4×F5, C22×F5, C2×C4×C12, C6×F5, C2×C4×F5, C12×F5, C2×C6×F5, F5×C2×C12
►On 120 points
Generators in S120
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 120)(14 109)(15 110)(16 111)(17 112)(18 113)(19 114)(20 115)(21 116)(22 117)(23 118)(24 119)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(49 101)(50 102)(51 103)(52 104)(53 105)(54 106)(55 107)(56 108)(57 97)(58 98)(59 99)(60 100)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(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)
(1 119 63 56 73)(2 120 64 57 74)(3 109 65 58 75)(4 110 66 59 76)(5 111 67 60 77)(6 112 68 49 78)(7 113 69 50 79)(8 114 70 51 80)(9 115 71 52 81)(10 116 72 53 82)(11 117 61 54 83)(12 118 62 55 84)(13 34 97 86 37)(14 35 98 87 38)(15 36 99 88 39)(16 25 100 89 40)(17 26 101 90 41)(18 27 102 91 42)(19 28 103 92 43)(20 29 104 93 44)(21 30 105 94 45)(22 31 106 95 46)(23 32 107 96 47)(24 33 108 85 48)
(1 48)(2 37)(3 38)(4 39)(5 40)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 64 86 57)(14 65 87 58)(15 66 88 59)(16 67 89 60)(17 68 90 49)(18 69 91 50)(19 70 92 51)(20 71 93 52)(21 72 94 53)(22 61 95 54)(23 62 96 55)(24 63 85 56)(25 77 100 111)(26 78 101 112)(27 79 102 113)(28 80 103 114)(29 81 104 115)(30 82 105 116)(31 83 106 117)(32 84 107 118)(33 73 108 119)(34 74 97 120)(35 75 98 109)(36 76 99 110)
G:=sub<Sym(120)| (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,120)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,119,63,56,73)(2,120,64,57,74)(3,109,65,58,75)(4,110,66,59,76)(5,111,67,60,77)(6,112,68,49,78)(7,113,69,50,79)(8,114,70,51,80)(9,115,71,52,81)(10,116,72,53,82)(11,117,61,54,83)(12,118,62,55,84)(13,34,97,86,37)(14,35,98,87,38)(15,36,99,88,39)(16,25,100,89,40)(17,26,101,90,41)(18,27,102,91,42)(19,28,103,92,43)(20,29,104,93,44)(21,30,105,94,45)(22,31,106,95,46)(23,32,107,96,47)(24,33,108,85,48), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,64,86,57)(14,65,87,58)(15,66,88,59)(16,67,89,60)(17,68,90,49)(18,69,91,50)(19,70,92,51)(20,71,93,52)(21,72,94,53)(22,61,95,54)(23,62,96,55)(24,63,85,56)(25,77,100,111)(26,78,101,112)(27,79,102,113)(28,80,103,114)(29,81,104,115)(30,82,105,116)(31,83,106,117)(32,84,107,118)(33,73,108,119)(34,74,97,120)(35,75,98,109)(36,76,99,110)>;
G:=Group( (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,120)(14,109)(15,110)(16,111)(17,112)(18,113)(19,114)(20,115)(21,116)(22,117)(23,118)(24,119)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,61)(32,62)(33,63)(34,64)(35,65)(36,66)(49,101)(50,102)(51,103)(52,104)(53,105)(54,106)(55,107)(56,108)(57,97)(58,98)(59,99)(60,100)(73,85)(74,86)(75,87)(76,88)(77,89)(78,90)(79,91)(80,92)(81,93)(82,94)(83,95)(84,96), (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), (1,119,63,56,73)(2,120,64,57,74)(3,109,65,58,75)(4,110,66,59,76)(5,111,67,60,77)(6,112,68,49,78)(7,113,69,50,79)(8,114,70,51,80)(9,115,71,52,81)(10,116,72,53,82)(11,117,61,54,83)(12,118,62,55,84)(13,34,97,86,37)(14,35,98,87,38)(15,36,99,88,39)(16,25,100,89,40)(17,26,101,90,41)(18,27,102,91,42)(19,28,103,92,43)(20,29,104,93,44)(21,30,105,94,45)(22,31,106,95,46)(23,32,107,96,47)(24,33,108,85,48), (1,48)(2,37)(3,38)(4,39)(5,40)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,64,86,57)(14,65,87,58)(15,66,88,59)(16,67,89,60)(17,68,90,49)(18,69,91,50)(19,70,92,51)(20,71,93,52)(21,72,94,53)(22,61,95,54)(23,62,96,55)(24,63,85,56)(25,77,100,111)(26,78,101,112)(27,79,102,113)(28,80,103,114)(29,81,104,115)(30,82,105,116)(31,83,106,117)(32,84,107,118)(33,73,108,119)(34,74,97,120)(35,75,98,109)(36,76,99,110) );
G=PermutationGroup([[(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,120),(14,109),(15,110),(16,111),(17,112),(18,113),(19,114),(20,115),(21,116),(22,117),(23,118),(24,119),(25,67),(26,68),(27,69),(28,70),(29,71),(30,72),(31,61),(32,62),(33,63),(34,64),(35,65),(36,66),(49,101),(50,102),(51,103),(52,104),(53,105),(54,106),(55,107),(56,108),(57,97),(58,98),(59,99),(60,100),(73,85),(74,86),(75,87),(76,88),(77,89),(78,90),(79,91),(80,92),(81,93),(82,94),(83,95),(84,96)], [(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)], [(1,119,63,56,73),(2,120,64,57,74),(3,109,65,58,75),(4,110,66,59,76),(5,111,67,60,77),(6,112,68,49,78),(7,113,69,50,79),(8,114,70,51,80),(9,115,71,52,81),(10,116,72,53,82),(11,117,61,54,83),(12,118,62,55,84),(13,34,97,86,37),(14,35,98,87,38),(15,36,99,88,39),(16,25,100,89,40),(17,26,101,90,41),(18,27,102,91,42),(19,28,103,92,43),(20,29,104,93,44),(21,30,105,94,45),(22,31,106,95,46),(23,32,107,96,47),(24,33,108,85,48)], [(1,48),(2,37),(3,38),(4,39),(5,40),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,64,86,57),(14,65,87,58),(15,66,88,59),(16,67,89,60),(17,68,90,49),(18,69,91,50),(19,70,92,51),(20,71,93,52),(21,72,94,53),(22,61,95,54),(23,62,96,55),(24,63,85,56),(25,77,100,111),(26,78,101,112),(27,79,102,113),(28,80,103,114),(29,81,104,115),(30,82,105,116),(31,83,106,117),(32,84,107,118),(33,73,108,119),(34,74,97,120),(35,75,98,109),(36,76,99,110)]])
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 |
Matrix representation of F5×C2×C12 ►in 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] >;
F5×C2×C12 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