metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.Dic5, C20.33D6, C15⋊6M4(2), C12.33D10, Dic3.Dic5, C60.33C22, C5⋊2C8⋊4S3, C5⋊5(C8⋊S3), (C4×S3).2D5, C15⋊3C8⋊12C2, C4.26(S3×D5), (S3×C20).3C2, (S3×C10).4C4, C10.18(C4×S3), C30.26(C2×C4), C3⋊1(C4.Dic5), C2.3(S3×Dic5), C6.2(C2×Dic5), (C5×Dic3).4C4, (C3×C5⋊2C8)⋊6C2, SmallGroup(240,11)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.Dic5
G = < a,b,c,d | a6=b2=1, c10=a3, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c9 >
Smallest permutation representation of D6.Dic5
►On 120 points
Generators in S120
(1 53 85 11 43 95)(2 54 86 12 44 96)(3 55 87 13 45 97)(4 56 88 14 46 98)(5 57 89 15 47 99)(6 58 90 16 48 100)(7 59 91 17 49 81)(8 60 92 18 50 82)(9 41 93 19 51 83)(10 42 94 20 52 84)(21 119 64 31 109 74)(22 120 65 32 110 75)(23 101 66 33 111 76)(24 102 67 34 112 77)(25 103 68 35 113 78)(26 104 69 36 114 79)(27 105 70 37 115 80)(28 106 71 38 116 61)(29 107 72 39 117 62)(30 108 73 40 118 63)
(1 95)(2 96)(3 97)(4 98)(5 99)(6 100)(7 81)(8 82)(9 83)(10 84)(11 85)(12 86)(13 87)(14 88)(15 89)(16 90)(17 91)(18 92)(19 93)(20 94)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 106)(62 107)(63 108)(64 109)(65 110)(66 111)(67 112)(68 113)(69 114)(70 115)(71 116)(72 117)(73 118)(74 119)(75 120)(76 101)(77 102)(78 103)(79 104)(80 105)
(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 73 6 78 11 63 16 68)(2 62 7 67 12 72 17 77)(3 71 8 76 13 61 18 66)(4 80 9 65 14 70 19 75)(5 69 10 74 15 79 20 64)(21 47 26 52 31 57 36 42)(22 56 27 41 32 46 37 51)(23 45 28 50 33 55 38 60)(24 54 29 59 34 44 39 49)(25 43 30 48 35 53 40 58)(81 102 86 107 91 112 96 117)(82 111 87 116 92 101 97 106)(83 120 88 105 93 110 98 115)(84 109 89 114 94 119 99 104)(85 118 90 103 95 108 100 113)
G:=sub<Sym(120)| (1,53,85,11,43,95)(2,54,86,12,44,96)(3,55,87,13,45,97)(4,56,88,14,46,98)(5,57,89,15,47,99)(6,58,90,16,48,100)(7,59,91,17,49,81)(8,60,92,18,50,82)(9,41,93,19,51,83)(10,42,94,20,52,84)(21,119,64,31,109,74)(22,120,65,32,110,75)(23,101,66,33,111,76)(24,102,67,34,112,77)(25,103,68,35,113,78)(26,104,69,36,114,79)(27,105,70,37,115,80)(28,106,71,38,116,61)(29,107,72,39,117,62)(30,108,73,40,118,63), (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,101)(77,102)(78,103)(79,104)(80,105), (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), (1,73,6,78,11,63,16,68)(2,62,7,67,12,72,17,77)(3,71,8,76,13,61,18,66)(4,80,9,65,14,70,19,75)(5,69,10,74,15,79,20,64)(21,47,26,52,31,57,36,42)(22,56,27,41,32,46,37,51)(23,45,28,50,33,55,38,60)(24,54,29,59,34,44,39,49)(25,43,30,48,35,53,40,58)(81,102,86,107,91,112,96,117)(82,111,87,116,92,101,97,106)(83,120,88,105,93,110,98,115)(84,109,89,114,94,119,99,104)(85,118,90,103,95,108,100,113)>;
G:=Group( (1,53,85,11,43,95)(2,54,86,12,44,96)(3,55,87,13,45,97)(4,56,88,14,46,98)(5,57,89,15,47,99)(6,58,90,16,48,100)(7,59,91,17,49,81)(8,60,92,18,50,82)(9,41,93,19,51,83)(10,42,94,20,52,84)(21,119,64,31,109,74)(22,120,65,32,110,75)(23,101,66,33,111,76)(24,102,67,34,112,77)(25,103,68,35,113,78)(26,104,69,36,114,79)(27,105,70,37,115,80)(28,106,71,38,116,61)(29,107,72,39,117,62)(30,108,73,40,118,63), (1,95)(2,96)(3,97)(4,98)(5,99)(6,100)(7,81)(8,82)(9,83)(10,84)(11,85)(12,86)(13,87)(14,88)(15,89)(16,90)(17,91)(18,92)(19,93)(20,94)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,106)(62,107)(63,108)(64,109)(65,110)(66,111)(67,112)(68,113)(69,114)(70,115)(71,116)(72,117)(73,118)(74,119)(75,120)(76,101)(77,102)(78,103)(79,104)(80,105), (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), (1,73,6,78,11,63,16,68)(2,62,7,67,12,72,17,77)(3,71,8,76,13,61,18,66)(4,80,9,65,14,70,19,75)(5,69,10,74,15,79,20,64)(21,47,26,52,31,57,36,42)(22,56,27,41,32,46,37,51)(23,45,28,50,33,55,38,60)(24,54,29,59,34,44,39,49)(25,43,30,48,35,53,40,58)(81,102,86,107,91,112,96,117)(82,111,87,116,92,101,97,106)(83,120,88,105,93,110,98,115)(84,109,89,114,94,119,99,104)(85,118,90,103,95,108,100,113) );
G=PermutationGroup([[(1,53,85,11,43,95),(2,54,86,12,44,96),(3,55,87,13,45,97),(4,56,88,14,46,98),(5,57,89,15,47,99),(6,58,90,16,48,100),(7,59,91,17,49,81),(8,60,92,18,50,82),(9,41,93,19,51,83),(10,42,94,20,52,84),(21,119,64,31,109,74),(22,120,65,32,110,75),(23,101,66,33,111,76),(24,102,67,34,112,77),(25,103,68,35,113,78),(26,104,69,36,114,79),(27,105,70,37,115,80),(28,106,71,38,116,61),(29,107,72,39,117,62),(30,108,73,40,118,63)], [(1,95),(2,96),(3,97),(4,98),(5,99),(6,100),(7,81),(8,82),(9,83),(10,84),(11,85),(12,86),(13,87),(14,88),(15,89),(16,90),(17,91),(18,92),(19,93),(20,94),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,106),(62,107),(63,108),(64,109),(65,110),(66,111),(67,112),(68,113),(69,114),(70,115),(71,116),(72,117),(73,118),(74,119),(75,120),(76,101),(77,102),(78,103),(79,104),(80,105)], [(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)], [(1,73,6,78,11,63,16,68),(2,62,7,67,12,72,17,77),(3,71,8,76,13,61,18,66),(4,80,9,65,14,70,19,75),(5,69,10,74,15,79,20,64),(21,47,26,52,31,57,36,42),(22,56,27,41,32,46,37,51),(23,45,28,50,33,55,38,60),(24,54,29,59,34,44,39,49),(25,43,30,48,35,53,40,58),(81,102,86,107,91,112,96,117),(82,111,87,116,92,101,97,106),(83,120,88,105,93,110,98,115),(84,109,89,114,94,119,99,104),(85,118,90,103,95,108,100,113)]])
D6.Dic5 is a maximal subgroup of
D5×C8⋊S3 C40⋊D6 C40.54D6 C40.55D6 D12.2Dic5 S3×C4.Dic5 D12.Dic5 D60.C22 C60.10C23 D20⋊10D6 D12.9D10 D12⋊D10 Dic10.26D6 D20.28D6 C60.44C23
D6.Dic5 is a maximal quotient of
C30.22C42 C60.94D4 C60.15Q8
42 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 6 | 2 | 1 | 1 | 6 | 2 | 2 | 2 | 10 | 10 | 30 | 30 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D5 | D6 | M4(2) | Dic5 | D10 | Dic5 | C4×S3 | C8⋊S3 | C4.Dic5 | S3×D5 | S3×Dic5 | D6.Dic5 |
| kernel | D6.Dic5 | C3×C5⋊2C8 | C15⋊3C8 | S3×C20 | C5×Dic3 | S3×C10 | C5⋊2C8 | C4×S3 | C20 | C15 | Dic3 | C12 | D6 | C10 | C5 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 4 |
Matrix representation of D6.Dic5 ►in GL4(𝔽241) generated by
| 1 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 1 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 208 | 1 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 79 | 6 |
| 78 | 156 | 0 | 0 |
| 85 | 163 | 0 | 0 |
| 0 | 0 | 94 | 228 |
| 0 | 0 | 63 | 147 |
G:=sub<GL(4,GF(241))| [1,240,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,1,240,0,0,0,0,240,208,0,0,0,1],[64,0,0,0,0,64,0,0,0,0,40,79,0,0,0,6],[78,85,0,0,156,163,0,0,0,0,94,63,0,0,228,147] >;
D6.Dic5 in GAP, Magma, Sage, TeX
D_6.{\rm Dic}_5 % in TeX
G:=Group("D6.Dic5"); // GroupNames label
G:=SmallGroup(240,11);
// by ID
G=gap.SmallGroup(240,11);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,121,50,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=1,c^10=a^3,d^2=c^5,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^9>;
// generators/relations
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