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, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, Q8, D5, C10, Dic3, C12, C12, C2×C6, C15, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C20, F5, D10, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C3×D5, C30, Q8⋊C4, C5⋊C8, Dic10, 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, C4, C22, S3, C2×C4, D4, 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 21 7 15)(2 20 8 14)(3 19 9 13)(4 18 10 24)(5 17 11 23)(6 16 12 22)(25 79 31 73)(26 78 32 84)(27 77 33 83)(28 76 34 82)(29 75 35 81)(30 74 36 80)(37 88 43 94)(38 87 44 93)(39 86 45 92)(40 85 46 91)(41 96 47 90)(42 95 48 89)(49 120 55 114)(50 119 56 113)(51 118 57 112)(52 117 58 111)(53 116 59 110)(54 115 60 109)(61 104 67 98)(62 103 68 97)(63 102 69 108)(64 101 70 107)(65 100 71 106)(66 99 72 105)
(1 45 25 56 102)(2 46 26 57 103)(3 47 27 58 104)(4 48 28 59 105)(5 37 29 60 106)(6 38 30 49 107)(7 39 31 50 108)(8 40 32 51 97)(9 41 33 52 98)(10 42 34 53 99)(11 43 35 54 100)(12 44 36 55 101)(13 96 83 117 61)(14 85 84 118 62)(15 86 73 119 63)(16 87 74 120 64)(17 88 75 109 65)(18 89 76 110 66)(19 90 77 111 67)(20 91 78 112 68)(21 92 79 113 69)(22 93 80 114 70)(23 94 81 115 71)(24 95 82 116 72)
(2 8)(4 10)(6 12)(13 22)(14 17)(15 24)(16 19)(18 21)(20 23)(25 102 56 45)(26 97 57 40)(27 104 58 47)(28 99 59 42)(29 106 60 37)(30 101 49 44)(31 108 50 39)(32 103 51 46)(33 98 52 41)(34 105 53 48)(35 100 54 43)(36 107 55 38)(61 114 96 80)(62 109 85 75)(63 116 86 82)(64 111 87 77)(65 118 88 84)(66 113 89 79)(67 120 90 74)(68 115 91 81)(69 110 92 76)(70 117 93 83)(71 112 94 78)(72 119 95 73)
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), (1,21,7,15)(2,20,8,14)(3,19,9,13)(4,18,10,24)(5,17,11,23)(6,16,12,22)(25,79,31,73)(26,78,32,84)(27,77,33,83)(28,76,34,82)(29,75,35,81)(30,74,36,80)(37,88,43,94)(38,87,44,93)(39,86,45,92)(40,85,46,91)(41,96,47,90)(42,95,48,89)(49,120,55,114)(50,119,56,113)(51,118,57,112)(52,117,58,111)(53,116,59,110)(54,115,60,109)(61,104,67,98)(62,103,68,97)(63,102,69,108)(64,101,70,107)(65,100,71,106)(66,99,72,105), (1,45,25,56,102)(2,46,26,57,103)(3,47,27,58,104)(4,48,28,59,105)(5,37,29,60,106)(6,38,30,49,107)(7,39,31,50,108)(8,40,32,51,97)(9,41,33,52,98)(10,42,34,53,99)(11,43,35,54,100)(12,44,36,55,101)(13,96,83,117,61)(14,85,84,118,62)(15,86,73,119,63)(16,87,74,120,64)(17,88,75,109,65)(18,89,76,110,66)(19,90,77,111,67)(20,91,78,112,68)(21,92,79,113,69)(22,93,80,114,70)(23,94,81,115,71)(24,95,82,116,72), (2,8)(4,10)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)(25,102,56,45)(26,97,57,40)(27,104,58,47)(28,99,59,42)(29,106,60,37)(30,101,49,44)(31,108,50,39)(32,103,51,46)(33,98,52,41)(34,105,53,48)(35,100,54,43)(36,107,55,38)(61,114,96,80)(62,109,85,75)(63,116,86,82)(64,111,87,77)(65,118,88,84)(66,113,89,79)(67,120,90,74)(68,115,91,81)(69,110,92,76)(70,117,93,83)(71,112,94,78)(72,119,95,73)>;
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), (1,21,7,15)(2,20,8,14)(3,19,9,13)(4,18,10,24)(5,17,11,23)(6,16,12,22)(25,79,31,73)(26,78,32,84)(27,77,33,83)(28,76,34,82)(29,75,35,81)(30,74,36,80)(37,88,43,94)(38,87,44,93)(39,86,45,92)(40,85,46,91)(41,96,47,90)(42,95,48,89)(49,120,55,114)(50,119,56,113)(51,118,57,112)(52,117,58,111)(53,116,59,110)(54,115,60,109)(61,104,67,98)(62,103,68,97)(63,102,69,108)(64,101,70,107)(65,100,71,106)(66,99,72,105), (1,45,25,56,102)(2,46,26,57,103)(3,47,27,58,104)(4,48,28,59,105)(5,37,29,60,106)(6,38,30,49,107)(7,39,31,50,108)(8,40,32,51,97)(9,41,33,52,98)(10,42,34,53,99)(11,43,35,54,100)(12,44,36,55,101)(13,96,83,117,61)(14,85,84,118,62)(15,86,73,119,63)(16,87,74,120,64)(17,88,75,109,65)(18,89,76,110,66)(19,90,77,111,67)(20,91,78,112,68)(21,92,79,113,69)(22,93,80,114,70)(23,94,81,115,71)(24,95,82,116,72), (2,8)(4,10)(6,12)(13,22)(14,17)(15,24)(16,19)(18,21)(20,23)(25,102,56,45)(26,97,57,40)(27,104,58,47)(28,99,59,42)(29,106,60,37)(30,101,49,44)(31,108,50,39)(32,103,51,46)(33,98,52,41)(34,105,53,48)(35,100,54,43)(36,107,55,38)(61,114,96,80)(62,109,85,75)(63,116,86,82)(64,111,87,77)(65,118,88,84)(66,113,89,79)(67,120,90,74)(68,115,91,81)(69,110,92,76)(70,117,93,83)(71,112,94,78)(72,119,95,73) );
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)], [(1,21,7,15),(2,20,8,14),(3,19,9,13),(4,18,10,24),(5,17,11,23),(6,16,12,22),(25,79,31,73),(26,78,32,84),(27,77,33,83),(28,76,34,82),(29,75,35,81),(30,74,36,80),(37,88,43,94),(38,87,44,93),(39,86,45,92),(40,85,46,91),(41,96,47,90),(42,95,48,89),(49,120,55,114),(50,119,56,113),(51,118,57,112),(52,117,58,111),(53,116,59,110),(54,115,60,109),(61,104,67,98),(62,103,68,97),(63,102,69,108),(64,101,70,107),(65,100,71,106),(66,99,72,105)], [(1,45,25,56,102),(2,46,26,57,103),(3,47,27,58,104),(4,48,28,59,105),(5,37,29,60,106),(6,38,30,49,107),(7,39,31,50,108),(8,40,32,51,97),(9,41,33,52,98),(10,42,34,53,99),(11,43,35,54,100),(12,44,36,55,101),(13,96,83,117,61),(14,85,84,118,62),(15,86,73,119,63),(16,87,74,120,64),(17,88,75,109,65),(18,89,76,110,66),(19,90,77,111,67),(20,91,78,112,68),(21,92,79,113,69),(22,93,80,114,70),(23,94,81,115,71),(24,95,82,116,72)], [(2,8),(4,10),(6,12),(13,22),(14,17),(15,24),(16,19),(18,21),(20,23),(25,102,56,45),(26,97,57,40),(27,104,58,47),(28,99,59,42),(29,106,60,37),(30,101,49,44),(31,108,50,39),(32,103,51,46),(33,98,52,41),(34,105,53,48),(35,100,54,43),(36,107,55,38),(61,114,96,80),(62,109,85,75),(63,116,86,82),(64,111,87,77),(65,118,88,84),(66,113,89,79),(67,120,90,74),(68,115,91,81),(69,110,92,76),(70,117,93,83),(71,112,94,78),(72,119,95,73)]])
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