metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊3F5, Dic30⋊1C4, Dic5.2D12, C4⋊F5.1S3, C4.9(S3×F5), C20.3(C4×S3), C60.2(C2×C4), C5⋊(C6.SD16), C12.2(C2×F5), (C5×Dic6)⋊1C4, (C6×D5).19D4, C3⋊2(Q8⋊F5), (C4×D5).21D6, C2.6(D6⋊F5), (C3×D5).1Q16, C10.3(D6⋊C4), C15⋊1(Q8⋊C4), (D5×Dic6).3C2, (C3×D5).3SD16, C6.3(C22⋊F5), C60.C4.1C2, C30.3(C22⋊C4), D5.2(D4.S3), (C3×Dic5).22D4, D5.2(C3⋊Q16), D10.23(C3⋊D4), (D5×C12).31C22, (C3×C4⋊F5).1C2, SmallGroup(480,229)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊F5
G = < a,b,c,d | a12=c5=d4=1, b2=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, dbd-1=a3b, dcd-1=c3 >
Subgroups: 468 in 84 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×4], C22, C5, C6, C6 [×2], C8, C2×C4 [×3], Q8 [×3], D5 [×2], C10, Dic3 [×2], C12, C12 [×2], C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C3⋊C8, Dic6, Dic6 [×2], C2×Dic3, C2×C12 [×2], C3×D5 [×2], C30, Q8⋊C4, C5⋊C8, Dic10 [×2], C4×D5, C4×D5, C5×Q8, C2×F5, C2×C3⋊C8, C3×C4⋊C4, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C3×F5, C6×D5, D5⋊C8, C4⋊F5, Q8×D5, C6.SD16, C15⋊C8, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, C6×F5, Q8⋊F5, C3×C4⋊F5, C60.C4, D5×Dic6, Dic6⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, SD16, Q16, F5, C4×S3, D12, C3⋊D4, Q8⋊C4, C2×F5, D6⋊C4, D4.S3, C3⋊Q16, C22⋊F5, C6.SD16, S3×F5, Q8⋊F5, D6⋊F5, Dic6⋊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 40 7 46)(2 39 8 45)(3 38 9 44)(4 37 10 43)(5 48 11 42)(6 47 12 41)(13 93 19 87)(14 92 20 86)(15 91 21 85)(16 90 22 96)(17 89 23 95)(18 88 24 94)(25 55 31 49)(26 54 32 60)(27 53 33 59)(28 52 34 58)(29 51 35 57)(30 50 36 56)(61 117 67 111)(62 116 68 110)(63 115 69 109)(64 114 70 120)(65 113 71 119)(66 112 72 118)(73 107 79 101)(74 106 80 100)(75 105 81 99)(76 104 82 98)(77 103 83 97)(78 102 84 108)
(1 85 57 109 74)(2 86 58 110 75)(3 87 59 111 76)(4 88 60 112 77)(5 89 49 113 78)(6 90 50 114 79)(7 91 51 115 80)(8 92 52 116 81)(9 93 53 117 82)(10 94 54 118 83)(11 95 55 119 84)(12 96 56 120 73)(13 27 61 104 38)(14 28 62 105 39)(15 29 63 106 40)(16 30 64 107 41)(17 31 65 108 42)(18 32 66 97 43)(19 33 67 98 44)(20 34 68 99 45)(21 35 69 100 46)(22 36 70 101 47)(23 25 71 102 48)(24 26 72 103 37)
(2 8)(4 10)(6 12)(13 36 104 70)(14 31 105 65)(15 26 106 72)(16 33 107 67)(17 28 108 62)(18 35 97 69)(19 30 98 64)(20 25 99 71)(21 32 100 66)(22 27 101 61)(23 34 102 68)(24 29 103 63)(37 40)(38 47)(39 42)(41 44)(43 46)(45 48)(49 78 113 89)(50 73 114 96)(51 80 115 91)(52 75 116 86)(53 82 117 93)(54 77 118 88)(55 84 119 95)(56 79 120 90)(57 74 109 85)(58 81 110 92)(59 76 111 87)(60 83 112 94)
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), (1,40,7,46)(2,39,8,45)(3,38,9,44)(4,37,10,43)(5,48,11,42)(6,47,12,41)(13,93,19,87)(14,92,20,86)(15,91,21,85)(16,90,22,96)(17,89,23,95)(18,88,24,94)(25,55,31,49)(26,54,32,60)(27,53,33,59)(28,52,34,58)(29,51,35,57)(30,50,36,56)(61,117,67,111)(62,116,68,110)(63,115,69,109)(64,114,70,120)(65,113,71,119)(66,112,72,118)(73,107,79,101)(74,106,80,100)(75,105,81,99)(76,104,82,98)(77,103,83,97)(78,102,84,108), (1,85,57,109,74)(2,86,58,110,75)(3,87,59,111,76)(4,88,60,112,77)(5,89,49,113,78)(6,90,50,114,79)(7,91,51,115,80)(8,92,52,116,81)(9,93,53,117,82)(10,94,54,118,83)(11,95,55,119,84)(12,96,56,120,73)(13,27,61,104,38)(14,28,62,105,39)(15,29,63,106,40)(16,30,64,107,41)(17,31,65,108,42)(18,32,66,97,43)(19,33,67,98,44)(20,34,68,99,45)(21,35,69,100,46)(22,36,70,101,47)(23,25,71,102,48)(24,26,72,103,37), (2,8)(4,10)(6,12)(13,36,104,70)(14,31,105,65)(15,26,106,72)(16,33,107,67)(17,28,108,62)(18,35,97,69)(19,30,98,64)(20,25,99,71)(21,32,100,66)(22,27,101,61)(23,34,102,68)(24,29,103,63)(37,40)(38,47)(39,42)(41,44)(43,46)(45,48)(49,78,113,89)(50,73,114,96)(51,80,115,91)(52,75,116,86)(53,82,117,93)(54,77,118,88)(55,84,119,95)(56,79,120,90)(57,74,109,85)(58,81,110,92)(59,76,111,87)(60,83,112,94)>;
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), (1,40,7,46)(2,39,8,45)(3,38,9,44)(4,37,10,43)(5,48,11,42)(6,47,12,41)(13,93,19,87)(14,92,20,86)(15,91,21,85)(16,90,22,96)(17,89,23,95)(18,88,24,94)(25,55,31,49)(26,54,32,60)(27,53,33,59)(28,52,34,58)(29,51,35,57)(30,50,36,56)(61,117,67,111)(62,116,68,110)(63,115,69,109)(64,114,70,120)(65,113,71,119)(66,112,72,118)(73,107,79,101)(74,106,80,100)(75,105,81,99)(76,104,82,98)(77,103,83,97)(78,102,84,108), (1,85,57,109,74)(2,86,58,110,75)(3,87,59,111,76)(4,88,60,112,77)(5,89,49,113,78)(6,90,50,114,79)(7,91,51,115,80)(8,92,52,116,81)(9,93,53,117,82)(10,94,54,118,83)(11,95,55,119,84)(12,96,56,120,73)(13,27,61,104,38)(14,28,62,105,39)(15,29,63,106,40)(16,30,64,107,41)(17,31,65,108,42)(18,32,66,97,43)(19,33,67,98,44)(20,34,68,99,45)(21,35,69,100,46)(22,36,70,101,47)(23,25,71,102,48)(24,26,72,103,37), (2,8)(4,10)(6,12)(13,36,104,70)(14,31,105,65)(15,26,106,72)(16,33,107,67)(17,28,108,62)(18,35,97,69)(19,30,98,64)(20,25,99,71)(21,32,100,66)(22,27,101,61)(23,34,102,68)(24,29,103,63)(37,40)(38,47)(39,42)(41,44)(43,46)(45,48)(49,78,113,89)(50,73,114,96)(51,80,115,91)(52,75,116,86)(53,82,117,93)(54,77,118,88)(55,84,119,95)(56,79,120,90)(57,74,109,85)(58,81,110,92)(59,76,111,87)(60,83,112,94) );
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)], [(1,40,7,46),(2,39,8,45),(3,38,9,44),(4,37,10,43),(5,48,11,42),(6,47,12,41),(13,93,19,87),(14,92,20,86),(15,91,21,85),(16,90,22,96),(17,89,23,95),(18,88,24,94),(25,55,31,49),(26,54,32,60),(27,53,33,59),(28,52,34,58),(29,51,35,57),(30,50,36,56),(61,117,67,111),(62,116,68,110),(63,115,69,109),(64,114,70,120),(65,113,71,119),(66,112,72,118),(73,107,79,101),(74,106,80,100),(75,105,81,99),(76,104,82,98),(77,103,83,97),(78,102,84,108)], [(1,85,57,109,74),(2,86,58,110,75),(3,87,59,111,76),(4,88,60,112,77),(5,89,49,113,78),(6,90,50,114,79),(7,91,51,115,80),(8,92,52,116,81),(9,93,53,117,82),(10,94,54,118,83),(11,95,55,119,84),(12,96,56,120,73),(13,27,61,104,38),(14,28,62,105,39),(15,29,63,106,40),(16,30,64,107,41),(17,31,65,108,42),(18,32,66,97,43),(19,33,67,98,44),(20,34,68,99,45),(21,35,69,100,46),(22,36,70,101,47),(23,25,71,102,48),(24,26,72,103,37)], [(2,8),(4,10),(6,12),(13,36,104,70),(14,31,105,65),(15,26,106,72),(16,33,107,67),(17,28,108,62),(18,35,97,69),(19,30,98,64),(20,25,99,71),(21,32,100,66),(22,27,101,61),(23,34,102,68),(24,29,103,63),(37,40),(38,47),(39,42),(41,44),(43,46),(45,48),(49,78,113,89),(50,73,114,96),(51,80,115,91),(52,75,116,86),(53,82,117,93),(54,77,118,88),(55,84,119,95),(56,79,120,90),(57,74,109,85),(58,81,110,92),(59,76,111,87),(60,83,112,94)])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10 | 12A | 12B | ··· | 12F | 15 | 20A | 20B | 20C | 30 | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 10 | 12 | 20 | 20 | 60 | 4 | 2 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 20 | ··· | 20 | 8 | 8 | 24 | 24 | 8 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D4 | D4 | D6 | SD16 | Q16 | D12 | C4×S3 | C3⋊D4 | F5 | C2×F5 | D4.S3 | C3⋊Q16 | C22⋊F5 | S3×F5 | Q8⋊F5 | D6⋊F5 | Dic6⋊F5 |
| kernel | Dic6⋊F5 | C3×C4⋊F5 | C60.C4 | D5×Dic6 | C5×Dic6 | Dic30 | C4⋊F5 | C3×Dic5 | C6×D5 | C4×D5 | C3×D5 | C3×D5 | Dic5 | C20 | D10 | Dic6 | C12 | D5 | D5 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of Dic6⋊F5 ►in GL8(𝔽241)
| 240 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 96 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 146 | 0 | 0 | 0 | 0 | 0 | 0 |
| 137 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 59 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 167 | 182 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 117 | 0 | 234 | 234 |
| 0 | 0 | 0 | 0 | 7 | 124 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 7 | 124 | 7 |
| 0 | 0 | 0 | 0 | 234 | 234 | 0 | 117 |
| 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 | 240 | 240 | 240 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 122 | 177 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 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 | 240 | 240 | 240 | 240 |
G:=sub<GL(8,GF(241))| [240,96,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,15,10,0,0,0,0,0,0,0,225,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240],[0,137,0,0,0,0,0,0,146,0,0,0,0,0,0,0,0,0,59,167,0,0,0,0,0,0,34,182,0,0,0,0,0,0,0,0,117,7,0,234,0,0,0,0,0,124,7,234,0,0,0,0,234,7,124,0,0,0,0,0,234,0,7,117],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,240,0,1,0,0,0,0,0,240,0,0,1,0,0,0,0,240,0,0,0],[64,122,0,0,0,0,0,0,0,177,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,1,0,240] >;
Dic6⋊F5 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes F_5 % in TeX
G:=Group("Dic6:F5"); // GroupNames label
G:=SmallGroup(480,229);
// by ID
G=gap.SmallGroup(480,229);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,675,346,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^5=d^4=1,b^2=a^6,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^7,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^3>;
// generators/relations