direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×Dic6, C20.13D6, Dic30⋊9C2, D10.16D6, C12.26D10, C30.1C23, C60.19C22, Dic5.11D6, Dic3.1D10, Dic15.1C22, (C3×D5)⋊Q8, C3⋊2(Q8×D5), C15⋊Q8⋊1C2, C15⋊1(C2×Q8), C5⋊1(C2×Dic6), (C4×D5).1S3, C4.12(S3×D5), (C5×Dic6)⋊2C2, (D5×C12).1C2, C6.1(C22×D5), C10.1(C22×S3), (D5×Dic3).1C2, (C6×D5).9C22, (C5×Dic3).1C22, (C3×Dic5).11C22, C2.5(C2×S3×D5), SmallGroup(240,125)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×Dic6
G = < a,b,c,d | a5=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 288 in 76 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, Q8, D5, C10, Dic3, Dic3, C12, C12, C2×C6, C15, C2×Q8, Dic5, Dic5, C20, C20, D10, Dic6, Dic6, C2×Dic3, C2×C12, C3×D5, C30, Dic10, C4×D5, C4×D5, C5×Q8, C2×Dic6, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, Q8×D5, D5×Dic3, C15⋊Q8, D5×C12, C5×Dic6, Dic30, D5×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D5, D6, C2×Q8, D10, Dic6, C22×S3, C22×D5, C2×Dic6, S3×D5, Q8×D5, C2×S3×D5, D5×Dic6
►On 120 points
Generators in S120
(1 105 116 81 63)(2 106 117 82 64)(3 107 118 83 65)(4 108 119 84 66)(5 97 120 73 67)(6 98 109 74 68)(7 99 110 75 69)(8 100 111 76 70)(9 101 112 77 71)(10 102 113 78 72)(11 103 114 79 61)(12 104 115 80 62)(13 94 51 34 41)(14 95 52 35 42)(15 96 53 36 43)(16 85 54 25 44)(17 86 55 26 45)(18 87 56 27 46)(19 88 57 28 47)(20 89 58 29 48)(21 90 59 30 37)(22 91 60 31 38)(23 92 49 32 39)(24 93 50 33 40)
(1 69)(2 70)(3 71)(4 72)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 85)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(109 115)(110 116)(111 117)(112 118)(113 119)(114 120)
(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 59 7 53)(2 58 8 52)(3 57 9 51)(4 56 10 50)(5 55 11 49)(6 54 12 60)(13 83 19 77)(14 82 20 76)(15 81 21 75)(16 80 22 74)(17 79 23 73)(18 78 24 84)(25 104 31 98)(26 103 32 97)(27 102 33 108)(28 101 34 107)(29 100 35 106)(30 99 36 105)(37 110 43 116)(38 109 44 115)(39 120 45 114)(40 119 46 113)(41 118 47 112)(42 117 48 111)(61 92 67 86)(62 91 68 85)(63 90 69 96)(64 89 70 95)(65 88 71 94)(66 87 72 93)
G:=sub<Sym(120)| (1,105,116,81,63)(2,106,117,82,64)(3,107,118,83,65)(4,108,119,84,66)(5,97,120,73,67)(6,98,109,74,68)(7,99,110,75,69)(8,100,111,76,70)(9,101,112,77,71)(10,102,113,78,72)(11,103,114,79,61)(12,104,115,80,62)(13,94,51,34,41)(14,95,52,35,42)(15,96,53,36,43)(16,85,54,25,44)(17,86,55,26,45)(18,87,56,27,46)(19,88,57,28,47)(20,89,58,29,48)(21,90,59,30,37)(22,91,60,31,38)(23,92,49,32,39)(24,93,50,33,40), (1,69)(2,70)(3,71)(4,72)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,85)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120), (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), (1,59,7,53)(2,58,8,52)(3,57,9,51)(4,56,10,50)(5,55,11,49)(6,54,12,60)(13,83,19,77)(14,82,20,76)(15,81,21,75)(16,80,22,74)(17,79,23,73)(18,78,24,84)(25,104,31,98)(26,103,32,97)(27,102,33,108)(28,101,34,107)(29,100,35,106)(30,99,36,105)(37,110,43,116)(38,109,44,115)(39,120,45,114)(40,119,46,113)(41,118,47,112)(42,117,48,111)(61,92,67,86)(62,91,68,85)(63,90,69,96)(64,89,70,95)(65,88,71,94)(66,87,72,93)>;
G:=Group( (1,105,116,81,63)(2,106,117,82,64)(3,107,118,83,65)(4,108,119,84,66)(5,97,120,73,67)(6,98,109,74,68)(7,99,110,75,69)(8,100,111,76,70)(9,101,112,77,71)(10,102,113,78,72)(11,103,114,79,61)(12,104,115,80,62)(13,94,51,34,41)(14,95,52,35,42)(15,96,53,36,43)(16,85,54,25,44)(17,86,55,26,45)(18,87,56,27,46)(19,88,57,28,47)(20,89,58,29,48)(21,90,59,30,37)(22,91,60,31,38)(23,92,49,32,39)(24,93,50,33,40), (1,69)(2,70)(3,71)(4,72)(5,61)(6,62)(7,63)(8,64)(9,65)(10,66)(11,67)(12,68)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(21,36)(22,25)(23,26)(24,27)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,86)(50,87)(51,88)(52,89)(53,90)(54,91)(55,92)(56,93)(57,94)(58,95)(59,96)(60,85)(73,103)(74,104)(75,105)(76,106)(77,107)(78,108)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(109,115)(110,116)(111,117)(112,118)(113,119)(114,120), (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), (1,59,7,53)(2,58,8,52)(3,57,9,51)(4,56,10,50)(5,55,11,49)(6,54,12,60)(13,83,19,77)(14,82,20,76)(15,81,21,75)(16,80,22,74)(17,79,23,73)(18,78,24,84)(25,104,31,98)(26,103,32,97)(27,102,33,108)(28,101,34,107)(29,100,35,106)(30,99,36,105)(37,110,43,116)(38,109,44,115)(39,120,45,114)(40,119,46,113)(41,118,47,112)(42,117,48,111)(61,92,67,86)(62,91,68,85)(63,90,69,96)(64,89,70,95)(65,88,71,94)(66,87,72,93) );
G=PermutationGroup([[(1,105,116,81,63),(2,106,117,82,64),(3,107,118,83,65),(4,108,119,84,66),(5,97,120,73,67),(6,98,109,74,68),(7,99,110,75,69),(8,100,111,76,70),(9,101,112,77,71),(10,102,113,78,72),(11,103,114,79,61),(12,104,115,80,62),(13,94,51,34,41),(14,95,52,35,42),(15,96,53,36,43),(16,85,54,25,44),(17,86,55,26,45),(18,87,56,27,46),(19,88,57,28,47),(20,89,58,29,48),(21,90,59,30,37),(22,91,60,31,38),(23,92,49,32,39),(24,93,50,33,40)], [(1,69),(2,70),(3,71),(4,72),(5,61),(6,62),(7,63),(8,64),(9,65),(10,66),(11,67),(12,68),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(21,36),(22,25),(23,26),(24,27),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,86),(50,87),(51,88),(52,89),(53,90),(54,91),(55,92),(56,93),(57,94),(58,95),(59,96),(60,85),(73,103),(74,104),(75,105),(76,106),(77,107),(78,108),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(109,115),(110,116),(111,117),(112,118),(113,119),(114,120)], [(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)], [(1,59,7,53),(2,58,8,52),(3,57,9,51),(4,56,10,50),(5,55,11,49),(6,54,12,60),(13,83,19,77),(14,82,20,76),(15,81,21,75),(16,80,22,74),(17,79,23,73),(18,78,24,84),(25,104,31,98),(26,103,32,97),(27,102,33,108),(28,101,34,107),(29,100,35,106),(30,99,36,105),(37,110,43,116),(38,109,44,115),(39,120,45,114),(40,119,46,113),(41,118,47,112),(42,117,48,111),(61,92,67,86),(62,91,68,85),(63,90,69,96),(64,89,70,95),(65,88,71,94),(66,87,72,93)]])
D5×Dic6 is a maximal subgroup of
Dic6⋊F5 Dic30⋊C4 Dic60⋊C2 C24.2D10 C60.8C23 D20.13D6 Dic6⋊5F5 D20.38D6 D20.39D6 C15⋊2- 1+4 D20.29D6 S3×Q8×D5
D5×Dic6 is a maximal quotient of
Dic5⋊5Dic6 Dic3⋊Dic10 Dic15⋊Q8 Dic30⋊17C4 Dic5.1Dic6 Dic5.2Dic6 D10⋊Dic6 C60.67D4 C60.68D4 Dic5⋊Dic6 Dic5.7Dic6 D10⋊1Dic6 D10⋊2Dic6 Dic15.D4 D10⋊4Dic6 C60⋊Q8
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | 10B | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 2 | 2 | 6 | 6 | 10 | 30 | 30 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 10 | 10 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | D6 | D10 | D10 | Dic6 | S3×D5 | Q8×D5 | C2×S3×D5 | D5×Dic6 |
| kernel | D5×Dic6 | D5×Dic3 | C15⋊Q8 | D5×C12 | C5×Dic6 | Dic30 | C4×D5 | C3×D5 | Dic6 | Dic5 | C20 | D10 | Dic3 | C12 | D5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of D5×Dic6 ►in GL4(𝔽61) generated by
| 43 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 43 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 53 | 40 |
| 0 | 0 | 32 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 58 | 23 |
| 0 | 0 | 42 | 3 |
G:=sub<GL(4,GF(61))| [43,60,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,43,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,53,32,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,58,42,0,0,23,3] >;
D5×Dic6 in GAP, Magma, Sage, TeX
D_5\times {\rm Dic}_6 % in TeX
G:=Group("D5xDic6"); // GroupNames label
G:=SmallGroup(240,125);
// by ID
G=gap.SmallGroup(240,125);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,116,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations