metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic6⋊5F5, Dic30⋊3C4, C3⋊F5⋊Q8, C3⋊2(Q8×F5), C15⋊Q8⋊2C4, C15⋊2(C4×Q8), C4⋊F5.2S3, C60.5(C2×C4), (C2×F5).2D6, C4.12(S3×F5), D5.1(S3×Q8), C20.10(C4×S3), C5⋊(Dic6⋊C4), (C5×Dic6)⋊3C4, (C4×D5).32D6, C12.14(C2×F5), C6.3(C22×F5), Dic3⋊F5.2C2, C30.3(C22×C4), (D5×Dic6).4C2, Dic5.2(C4×S3), Dic3.2(C2×F5), (Dic3×F5).2C2, (C6×F5).2C22, Dic15.2(C2×C4), (C6×D5).23C23, D5.1(D4⋊2S3), D10.26(C22×S3), (D5×C12).34C22, (D5×Dic3).6C22, C2.8(C2×S3×F5), C10.3(S3×C2×C4), (C4×C3⋊F5).1C2, (C3×C4⋊F5).2C2, (C3×D5).2(C2×Q8), (C2×C3⋊F5).7C22, (C3×D5).3(C4○D4), (C5×Dic3).2(C2×C4), (C3×Dic5).21(C2×C4), SmallGroup(480,984)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊5F5
G = < a,b,c,d | a12=c5=d4=1, b2=a6, bab-1=a-1, ac=ca, dad-1=a7, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 660 in 140 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2 [×2], C3, C4, C4 [×10], C22, C5, C6, C6 [×2], C2×C4 [×7], Q8 [×4], D5 [×2], C10, Dic3 [×2], Dic3 [×5], C12, C12 [×3], C2×C6, C15, C42 [×3], C4⋊C4 [×3], C2×Q8, Dic5, Dic5 [×2], C20, C20 [×2], F5 [×5], D10, Dic6, Dic6 [×3], C2×Dic3 [×4], C2×C12 [×3], C3×D5 [×2], C30, C4×Q8, Dic10 [×3], C4×D5, C4×D5 [×2], C5×Q8, C2×F5 [×2], C2×F5 [×2], C4×Dic3 [×3], Dic3⋊C4 [×2], C3×C4⋊C4, C2×Dic6, C5×Dic3 [×2], C3×Dic5, Dic15 [×2], C60, C3×F5 [×2], C3⋊F5 [×2], C3⋊F5, C6×D5, C4×F5 [×3], C4⋊F5, C4⋊F5 [×2], Q8×D5, Dic6⋊C4, D5×Dic3 [×2], C15⋊Q8 [×2], D5×C12, C5×Dic6, Dic30, C6×F5 [×2], C2×C3⋊F5 [×2], Q8×F5, Dic3×F5 [×2], Dic3⋊F5 [×2], C3×C4⋊F5, C4×C3⋊F5, D5×Dic6, Dic6⋊5F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], Q8 [×2], C23, D6 [×3], C22×C4, C2×Q8, C4○D4, F5, C4×S3 [×2], C22×S3, C4×Q8, C2×F5 [×3], S3×C2×C4, D4⋊2S3, S3×Q8, C22×F5, Dic6⋊C4, S3×F5, Q8×F5, C2×S3×F5, Dic6⋊5F5
►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 42 7 48)(2 41 8 47)(3 40 9 46)(4 39 10 45)(5 38 11 44)(6 37 12 43)(13 52 19 58)(14 51 20 57)(15 50 21 56)(16 49 22 55)(17 60 23 54)(18 59 24 53)(25 112 31 118)(26 111 32 117)(27 110 33 116)(28 109 34 115)(29 120 35 114)(30 119 36 113)(61 107 67 101)(62 106 68 100)(63 105 69 99)(64 104 70 98)(65 103 71 97)(66 102 72 108)(73 90 79 96)(74 89 80 95)(75 88 81 94)(76 87 82 93)(77 86 83 92)(78 85 84 91)
(1 99 15 80 109)(2 100 16 81 110)(3 101 17 82 111)(4 102 18 83 112)(5 103 19 84 113)(6 104 20 73 114)(7 105 21 74 115)(8 106 22 75 116)(9 107 23 76 117)(10 108 24 77 118)(11 97 13 78 119)(12 98 14 79 120)(25 45 66 53 86)(26 46 67 54 87)(27 47 68 55 88)(28 48 69 56 89)(29 37 70 57 90)(30 38 71 58 91)(31 39 72 59 92)(32 40 61 60 93)(33 41 62 49 94)(34 42 63 50 95)(35 43 64 51 96)(36 44 65 52 85)
(1 48 7 42)(2 43 8 37)(3 38 9 44)(4 45 10 39)(5 40 11 46)(6 47 12 41)(13 26 84 61)(14 33 73 68)(15 28 74 63)(16 35 75 70)(17 30 76 65)(18 25 77 72)(19 32 78 67)(20 27 79 62)(21 34 80 69)(22 29 81 64)(23 36 82 71)(24 31 83 66)(49 114 88 98)(50 109 89 105)(51 116 90 100)(52 111 91 107)(53 118 92 102)(54 113 93 97)(55 120 94 104)(56 115 95 99)(57 110 96 106)(58 117 85 101)(59 112 86 108)(60 119 87 103)
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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,52,19,58)(14,51,20,57)(15,50,21,56)(16,49,22,55)(17,60,23,54)(18,59,24,53)(25,112,31,118)(26,111,32,117)(27,110,33,116)(28,109,34,115)(29,120,35,114)(30,119,36,113)(61,107,67,101)(62,106,68,100)(63,105,69,99)(64,104,70,98)(65,103,71,97)(66,102,72,108)(73,90,79,96)(74,89,80,95)(75,88,81,94)(76,87,82,93)(77,86,83,92)(78,85,84,91), (1,99,15,80,109)(2,100,16,81,110)(3,101,17,82,111)(4,102,18,83,112)(5,103,19,84,113)(6,104,20,73,114)(7,105,21,74,115)(8,106,22,75,116)(9,107,23,76,117)(10,108,24,77,118)(11,97,13,78,119)(12,98,14,79,120)(25,45,66,53,86)(26,46,67,54,87)(27,47,68,55,88)(28,48,69,56,89)(29,37,70,57,90)(30,38,71,58,91)(31,39,72,59,92)(32,40,61,60,93)(33,41,62,49,94)(34,42,63,50,95)(35,43,64,51,96)(36,44,65,52,85), (1,48,7,42)(2,43,8,37)(3,38,9,44)(4,45,10,39)(5,40,11,46)(6,47,12,41)(13,26,84,61)(14,33,73,68)(15,28,74,63)(16,35,75,70)(17,30,76,65)(18,25,77,72)(19,32,78,67)(20,27,79,62)(21,34,80,69)(22,29,81,64)(23,36,82,71)(24,31,83,66)(49,114,88,98)(50,109,89,105)(51,116,90,100)(52,111,91,107)(53,118,92,102)(54,113,93,97)(55,120,94,104)(56,115,95,99)(57,110,96,106)(58,117,85,101)(59,112,86,108)(60,119,87,103)>;
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), (1,42,7,48)(2,41,8,47)(3,40,9,46)(4,39,10,45)(5,38,11,44)(6,37,12,43)(13,52,19,58)(14,51,20,57)(15,50,21,56)(16,49,22,55)(17,60,23,54)(18,59,24,53)(25,112,31,118)(26,111,32,117)(27,110,33,116)(28,109,34,115)(29,120,35,114)(30,119,36,113)(61,107,67,101)(62,106,68,100)(63,105,69,99)(64,104,70,98)(65,103,71,97)(66,102,72,108)(73,90,79,96)(74,89,80,95)(75,88,81,94)(76,87,82,93)(77,86,83,92)(78,85,84,91), (1,99,15,80,109)(2,100,16,81,110)(3,101,17,82,111)(4,102,18,83,112)(5,103,19,84,113)(6,104,20,73,114)(7,105,21,74,115)(8,106,22,75,116)(9,107,23,76,117)(10,108,24,77,118)(11,97,13,78,119)(12,98,14,79,120)(25,45,66,53,86)(26,46,67,54,87)(27,47,68,55,88)(28,48,69,56,89)(29,37,70,57,90)(30,38,71,58,91)(31,39,72,59,92)(32,40,61,60,93)(33,41,62,49,94)(34,42,63,50,95)(35,43,64,51,96)(36,44,65,52,85), (1,48,7,42)(2,43,8,37)(3,38,9,44)(4,45,10,39)(5,40,11,46)(6,47,12,41)(13,26,84,61)(14,33,73,68)(15,28,74,63)(16,35,75,70)(17,30,76,65)(18,25,77,72)(19,32,78,67)(20,27,79,62)(21,34,80,69)(22,29,81,64)(23,36,82,71)(24,31,83,66)(49,114,88,98)(50,109,89,105)(51,116,90,100)(52,111,91,107)(53,118,92,102)(54,113,93,97)(55,120,94,104)(56,115,95,99)(57,110,96,106)(58,117,85,101)(59,112,86,108)(60,119,87,103) );
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)], [(1,42,7,48),(2,41,8,47),(3,40,9,46),(4,39,10,45),(5,38,11,44),(6,37,12,43),(13,52,19,58),(14,51,20,57),(15,50,21,56),(16,49,22,55),(17,60,23,54),(18,59,24,53),(25,112,31,118),(26,111,32,117),(27,110,33,116),(28,109,34,115),(29,120,35,114),(30,119,36,113),(61,107,67,101),(62,106,68,100),(63,105,69,99),(64,104,70,98),(65,103,71,97),(66,102,72,108),(73,90,79,96),(74,89,80,95),(75,88,81,94),(76,87,82,93),(77,86,83,92),(78,85,84,91)], [(1,99,15,80,109),(2,100,16,81,110),(3,101,17,82,111),(4,102,18,83,112),(5,103,19,84,113),(6,104,20,73,114),(7,105,21,74,115),(8,106,22,75,116),(9,107,23,76,117),(10,108,24,77,118),(11,97,13,78,119),(12,98,14,79,120),(25,45,66,53,86),(26,46,67,54,87),(27,47,68,55,88),(28,48,69,56,89),(29,37,70,57,90),(30,38,71,58,91),(31,39,72,59,92),(32,40,61,60,93),(33,41,62,49,94),(34,42,63,50,95),(35,43,64,51,96),(36,44,65,52,85)], [(1,48,7,42),(2,43,8,37),(3,38,9,44),(4,45,10,39),(5,40,11,46),(6,47,12,41),(13,26,84,61),(14,33,73,68),(15,28,74,63),(16,35,75,70),(17,30,76,65),(18,25,77,72),(19,32,78,67),(20,27,79,62),(21,34,80,69),(22,29,81,64),(23,36,82,71),(24,31,83,66),(49,114,88,98),(50,109,89,105),(51,116,90,100),(52,111,91,107),(53,118,92,102),(54,113,93,97),(55,120,94,104),(56,115,95,99),(57,110,96,106),(58,117,85,101),(59,112,86,108),(60,119,87,103)])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | ··· | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5 | 6A | 6B | 6C | 10 | 12A | 12B | ··· | 12F | 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 | 4 | 5 | 6 | 6 | 6 | 10 | 12 | 12 | ··· | 12 | 15 | 20 | 20 | 20 | 30 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 30 | 30 | 4 | 2 | 10 | 10 | 4 | 4 | 20 | ··· | 20 | 8 | 8 | 24 | 24 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 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 | F5 | C2×F5 | C2×F5 | D4⋊2S3 | S3×Q8 | S3×F5 | Q8×F5 | C2×S3×F5 | Dic6⋊5F5 |
| kernel | Dic6⋊5F5 | Dic3×F5 | Dic3⋊F5 | C3×C4⋊F5 | C4×C3⋊F5 | D5×Dic6 | C15⋊Q8 | C5×Dic6 | Dic30 | C4⋊F5 | C3⋊F5 | C4×D5 | C2×F5 | C3×D5 | Dic5 | C20 | Dic6 | Dic3 | C12 | D5 | D5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of Dic6⋊5F5 ►in GL8(𝔽61)
| 60 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 18 | 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 |
| 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 |
| 50 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(8,GF(61))| [60,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,43,38,0,0,0,0,0,0,38,18,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60],[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],[50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,15,9,0,0,0,0,0,0,9,46,0,0,0,0,0,0,0,0,60,0,0,1,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,1,0,0,0,0,0,60,0,1] >;
Dic6⋊5F5 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes_5F_5 % in TeX
G:=Group("Dic6:5F5"); // GroupNames label
G:=SmallGroup(480,984);
// by ID
G=gap.SmallGroup(480,984);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,120,422,219,100,1356,9414,2379]);
// 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,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations