direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: F5×Dic6, Dic30⋊6C4, (C3×F5)⋊Q8, C5⋊(C4×Dic6), C3⋊1(Q8×F5), C15⋊Q8⋊1C4, C15⋊1(C4×Q8), C4.5(S3×F5), C20.9(C4×S3), (C2×F5).9D6, (C4×F5).1S3, C60.12(C2×C4), (C5×Dic6)⋊6C4, (C4×D5).31D6, (C12×F5).1C2, C12.26(C2×F5), C60⋊C4.2C2, C6.1(C22×F5), Dic3⋊F5.1C2, C30.1(C22×C4), Dic3.1(C2×F5), (Dic3×F5).1C2, D5.1(C2×Dic6), Dic5.1(C4×S3), (D5×Dic6).6C2, (C6×F5).7C22, D5.1(C4○D12), Dic15.1(C2×C4), (C6×D5).21C23, D10.24(C22×S3), (D5×C12).42C22, (D5×Dic3).5C22, C2.6(C2×S3×F5), C10.1(S3×C2×C4), (C3×D5).1(C2×Q8), (C2×C3⋊F5).1C22, (C3×D5).1(C4○D4), (C5×Dic3).1(C2×C4), (C3×Dic5).19(C2×C4), SmallGroup(480,982)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for F5×Dic6
G = < a,b,c,d | a5=b4=c12=1, d2=c6, bab-1=a3, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 660 in 140 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, Q8, D5, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C42, C4⋊C4, C2×Q8, Dic5, Dic5, C20, C20, F5, F5, D10, Dic6, Dic6, C2×Dic3, C2×C12, C3×D5, C30, C4×Q8, Dic10, C4×D5, C4×D5, C5×Q8, C2×F5, C2×F5, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4×C12, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C3×F5, C3⋊F5, C6×D5, C4×F5, C4×F5, C4⋊F5, Q8×D5, C4×Dic6, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, C6×F5, C2×C3⋊F5, Q8×F5, Dic3×F5, Dic3⋊F5, C12×F5, C60⋊C4, D5×Dic6, F5×Dic6
Quotients: C1, C2, C4, C22, S3, C2×C4, Q8, C23, D6, C22×C4, C2×Q8, C4○D4, F5, Dic6, C4×S3, C22×S3, C4×Q8, C2×F5, C2×Dic6, S3×C2×C4, C4○D12, C22×F5, C4×Dic6, S3×F5, Q8×F5, C2×S3×F5, F5×Dic6
►On 120 points
Generators in S120
(1 61 49 109 78)(2 62 50 110 79)(3 63 51 111 80)(4 64 52 112 81)(5 65 53 113 82)(6 66 54 114 83)(7 67 55 115 84)(8 68 56 116 73)(9 69 57 117 74)(10 70 58 118 75)(11 71 59 119 76)(12 72 60 120 77)(13 39 29 99 95)(14 40 30 100 96)(15 41 31 101 85)(16 42 32 102 86)(17 43 33 103 87)(18 44 34 104 88)(19 45 35 105 89)(20 46 36 106 90)(21 47 25 107 91)(22 48 26 108 92)(23 37 27 97 93)(24 38 28 98 94)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 105 95 45)(14 106 96 46)(15 107 85 47)(16 108 86 48)(17 97 87 37)(18 98 88 38)(19 99 89 39)(20 100 90 40)(21 101 91 41)(22 102 92 42)(23 103 93 43)(24 104 94 44)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(49 84 109 67)(50 73 110 68)(51 74 111 69)(52 75 112 70)(53 76 113 71)(54 77 114 72)(55 78 115 61)(56 79 116 62)(57 80 117 63)(58 81 118 64)(59 82 119 65)(60 83 120 66)
(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 29 7 35)(2 28 8 34)(3 27 9 33)(4 26 10 32)(5 25 11 31)(6 36 12 30)(13 115 19 109)(14 114 20 120)(15 113 21 119)(16 112 22 118)(17 111 23 117)(18 110 24 116)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)(49 95 55 89)(50 94 56 88)(51 93 57 87)(52 92 58 86)(53 91 59 85)(54 90 60 96)(61 99 67 105)(62 98 68 104)(63 97 69 103)(64 108 70 102)(65 107 71 101)(66 106 72 100)
G:=sub<Sym(120)| (1,61,49,109,78)(2,62,50,110,79)(3,63,51,111,80)(4,64,52,112,81)(5,65,53,113,82)(6,66,54,114,83)(7,67,55,115,84)(8,68,56,116,73)(9,69,57,117,74)(10,70,58,118,75)(11,71,59,119,76)(12,72,60,120,77)(13,39,29,99,95)(14,40,30,100,96)(15,41,31,101,85)(16,42,32,102,86)(17,43,33,103,87)(18,44,34,104,88)(19,45,35,105,89)(20,46,36,106,90)(21,47,25,107,91)(22,48,26,108,92)(23,37,27,97,93)(24,38,28,98,94), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,105,95,45)(14,106,96,46)(15,107,85,47)(16,108,86,48)(17,97,87,37)(18,98,88,38)(19,99,89,39)(20,100,90,40)(21,101,91,41)(22,102,92,42)(23,103,93,43)(24,104,94,44)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,84,109,67)(50,73,110,68)(51,74,111,69)(52,75,112,70)(53,76,113,71)(54,77,114,72)(55,78,115,61)(56,79,116,62)(57,80,117,63)(58,81,118,64)(59,82,119,65)(60,83,120,66), (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), (1,29,7,35)(2,28,8,34)(3,27,9,33)(4,26,10,32)(5,25,11,31)(6,36,12,30)(13,115,19,109)(14,114,20,120)(15,113,21,119)(16,112,22,118)(17,111,23,117)(18,110,24,116)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,95,55,89)(50,94,56,88)(51,93,57,87)(52,92,58,86)(53,91,59,85)(54,90,60,96)(61,99,67,105)(62,98,68,104)(63,97,69,103)(64,108,70,102)(65,107,71,101)(66,106,72,100)>;
G:=Group( (1,61,49,109,78)(2,62,50,110,79)(3,63,51,111,80)(4,64,52,112,81)(5,65,53,113,82)(6,66,54,114,83)(7,67,55,115,84)(8,68,56,116,73)(9,69,57,117,74)(10,70,58,118,75)(11,71,59,119,76)(12,72,60,120,77)(13,39,29,99,95)(14,40,30,100,96)(15,41,31,101,85)(16,42,32,102,86)(17,43,33,103,87)(18,44,34,104,88)(19,45,35,105,89)(20,46,36,106,90)(21,47,25,107,91)(22,48,26,108,92)(23,37,27,97,93)(24,38,28,98,94), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,105,95,45)(14,106,96,46)(15,107,85,47)(16,108,86,48)(17,97,87,37)(18,98,88,38)(19,99,89,39)(20,100,90,40)(21,101,91,41)(22,102,92,42)(23,103,93,43)(24,104,94,44)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(49,84,109,67)(50,73,110,68)(51,74,111,69)(52,75,112,70)(53,76,113,71)(54,77,114,72)(55,78,115,61)(56,79,116,62)(57,80,117,63)(58,81,118,64)(59,82,119,65)(60,83,120,66), (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), (1,29,7,35)(2,28,8,34)(3,27,9,33)(4,26,10,32)(5,25,11,31)(6,36,12,30)(13,115,19,109)(14,114,20,120)(15,113,21,119)(16,112,22,118)(17,111,23,117)(18,110,24,116)(37,74,43,80)(38,73,44,79)(39,84,45,78)(40,83,46,77)(41,82,47,76)(42,81,48,75)(49,95,55,89)(50,94,56,88)(51,93,57,87)(52,92,58,86)(53,91,59,85)(54,90,60,96)(61,99,67,105)(62,98,68,104)(63,97,69,103)(64,108,70,102)(65,107,71,101)(66,106,72,100) );
G=PermutationGroup([[(1,61,49,109,78),(2,62,50,110,79),(3,63,51,111,80),(4,64,52,112,81),(5,65,53,113,82),(6,66,54,114,83),(7,67,55,115,84),(8,68,56,116,73),(9,69,57,117,74),(10,70,58,118,75),(11,71,59,119,76),(12,72,60,120,77),(13,39,29,99,95),(14,40,30,100,96),(15,41,31,101,85),(16,42,32,102,86),(17,43,33,103,87),(18,44,34,104,88),(19,45,35,105,89),(20,46,36,106,90),(21,47,25,107,91),(22,48,26,108,92),(23,37,27,97,93),(24,38,28,98,94)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,105,95,45),(14,106,96,46),(15,107,85,47),(16,108,86,48),(17,97,87,37),(18,98,88,38),(19,99,89,39),(20,100,90,40),(21,101,91,41),(22,102,92,42),(23,103,93,43),(24,104,94,44),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(49,84,109,67),(50,73,110,68),(51,74,111,69),(52,75,112,70),(53,76,113,71),(54,77,114,72),(55,78,115,61),(56,79,116,62),(57,80,117,63),(58,81,118,64),(59,82,119,65),(60,83,120,66)], [(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)], [(1,29,7,35),(2,28,8,34),(3,27,9,33),(4,26,10,32),(5,25,11,31),(6,36,12,30),(13,115,19,109),(14,114,20,120),(15,113,21,119),(16,112,22,118),(17,111,23,117),(18,110,24,116),(37,74,43,80),(38,73,44,79),(39,84,45,78),(40,83,46,77),(41,82,47,76),(42,81,48,75),(49,95,55,89),(50,94,56,88),(51,93,57,87),(52,92,58,86),(53,91,59,85),(54,90,60,96),(61,99,67,105),(62,98,68,104),(63,97,69,103),(64,108,70,102),(65,107,71,101),(66,106,72,100)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5 | 6A | 6B | 6C | 10 | 12A | 12B | 12C | ··· | 12L | 15 | 20A | 20B | 20C | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 10 | 12 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 10 | 30 | ··· | 30 | 4 | 2 | 10 | 10 | 4 | 2 | 2 | 10 | ··· | 10 | 8 | 8 | 24 | 24 | 8 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | - | + | + | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | Q8 | D6 | D6 | C4○D4 | C4×S3 | C4×S3 | Dic6 | C4○D12 | F5 | C2×F5 | C2×F5 | S3×F5 | Q8×F5 | C2×S3×F5 | F5×Dic6 |
| kernel | F5×Dic6 | Dic3×F5 | Dic3⋊F5 | C12×F5 | C60⋊C4 | D5×Dic6 | C15⋊Q8 | C5×Dic6 | Dic30 | C4×F5 | C3×F5 | C4×D5 | C2×F5 | C3×D5 | Dic5 | C20 | F5 | D5 | Dic6 | Dic3 | C12 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of F5×Dic6 ►in GL8(𝔽61)
| 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 | 60 | 60 | 60 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 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 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 60 | 60 | 60 |
| 60 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 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 | 0 | 0 | 0 | 0 | 1 |
| 44 | 55 | 0 | 0 | 0 | 0 | 0 | 0 |
| 28 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(8,GF(61))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,0,0,60,0,1,0,0,0,0,0,60,0,0,1,0,0,0,0,60,0,0,0],[60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,0,0,60,0,0,0,0,0,1,0,60],[60,40,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[44,28,0,0,0,0,0,0,55,17,0,0,0,0,0,0,0,0,34,24,0,0,0,0,0,0,51,27,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60] >;
F5×Dic6 in GAP, Magma, Sage, TeX
F_5\times {\rm Dic}_6 % in TeX
G:=Group("F5xDic6"); // GroupNames label
G:=SmallGroup(480,982);
// by ID
G=gap.SmallGroup(480,982);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,219,100,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^4=c^12=1,d^2=c^6,b*a*b^-1=a^3,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations