metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.10D20, C30.13D4, D10⋊1Dic3, (C6×D5)⋊1C4, C6.13(C4×D5), (C2×C6).6D10, (C2×C10).6D6, C15⋊4(C22⋊C4), C30.29(C2×C4), (C2×Dic3)⋊1D5, C2.4(D5×Dic3), C3⋊3(D10⋊C4), C22.5(S3×D5), (C10×Dic3)⋊1C2, (C2×Dic15)⋊5C2, C2.1(C15⋊D4), C5⋊2(C6.D4), C2.1(C3⋊D20), C6.11(C5⋊D4), (C2×C30).3C22, (C22×D5).2S3, C10.11(C3⋊D4), C10.11(C2×Dic3), (D5×C2×C6).1C2, SmallGroup(240,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊Dic3
G = < a,b,c,d | a10=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c-1 >
Subgroups: 272 in 68 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C6, C2×C4, C23, D5, C10, Dic3, C2×C6, C2×C6, C15, C22⋊C4, Dic5, C20, D10, D10, C2×C10, C2×Dic3, C2×Dic3, C22×C6, C3×D5, C30, C2×Dic5, C2×C20, C22×D5, C6.D4, C5×Dic3, Dic15, C6×D5, C6×D5, C2×C30, D10⋊C4, C10×Dic3, C2×Dic15, D5×C2×C6, D10⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, D5×Dic3, C15⋊D4, C3⋊D20, D10⋊Dic3
►On 120 points
(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 52)(2 51)(3 60)(4 59)(5 58)(6 57)(7 56)(8 55)(9 54)(10 53)(11 99)(12 98)(13 97)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 100)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 50)(29 49)(30 48)(31 68)(32 67)(33 66)(34 65)(35 64)(36 63)(37 62)(38 61)(39 70)(40 69)(71 103)(72 102)(73 101)(74 110)(75 109)(76 108)(77 107)(78 106)(79 105)(80 104)(81 119)(82 118)(83 117)(84 116)(85 115)(86 114)(87 113)(88 112)(89 111)(90 120)
(1 65 29 53 35 50)(2 66 30 54 36 41)(3 67 21 55 37 42)(4 68 22 56 38 43)(5 69 23 57 39 44)(6 70 24 58 40 45)(7 61 25 59 31 46)(8 62 26 60 32 47)(9 63 27 51 33 48)(10 64 28 52 34 49)(11 83 110 95 113 80)(12 84 101 96 114 71)(13 85 102 97 115 72)(14 86 103 98 116 73)(15 87 104 99 117 74)(16 88 105 100 118 75)(17 89 106 91 119 76)(18 90 107 92 120 77)(19 81 108 93 111 78)(20 82 109 94 112 79)
(1 113 53 83)(2 114 54 84)(3 115 55 85)(4 116 56 86)(5 117 57 87)(6 118 58 88)(7 119 59 89)(8 120 60 90)(9 111 51 81)(10 112 52 82)(11 65 95 35)(12 66 96 36)(13 67 97 37)(14 68 98 38)(15 69 99 39)(16 70 100 40)(17 61 91 31)(18 62 92 32)(19 63 93 33)(20 64 94 34)(21 102 42 72)(22 103 43 73)(23 104 44 74)(24 105 45 75)(25 106 46 76)(26 107 47 77)(27 108 48 78)(28 109 49 79)(29 110 50 80)(30 101 41 71)
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), (1,52)(2,51)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,100)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,50)(29,49)(30,48)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,70)(40,69)(71,103)(72,102)(73,101)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,120), (1,65,29,53,35,50)(2,66,30,54,36,41)(3,67,21,55,37,42)(4,68,22,56,38,43)(5,69,23,57,39,44)(6,70,24,58,40,45)(7,61,25,59,31,46)(8,62,26,60,32,47)(9,63,27,51,33,48)(10,64,28,52,34,49)(11,83,110,95,113,80)(12,84,101,96,114,71)(13,85,102,97,115,72)(14,86,103,98,116,73)(15,87,104,99,117,74)(16,88,105,100,118,75)(17,89,106,91,119,76)(18,90,107,92,120,77)(19,81,108,93,111,78)(20,82,109,94,112,79), (1,113,53,83)(2,114,54,84)(3,115,55,85)(4,116,56,86)(5,117,57,87)(6,118,58,88)(7,119,59,89)(8,120,60,90)(9,111,51,81)(10,112,52,82)(11,65,95,35)(12,66,96,36)(13,67,97,37)(14,68,98,38)(15,69,99,39)(16,70,100,40)(17,61,91,31)(18,62,92,32)(19,63,93,33)(20,64,94,34)(21,102,42,72)(22,103,43,73)(23,104,44,74)(24,105,45,75)(25,106,46,76)(26,107,47,77)(27,108,48,78)(28,109,49,79)(29,110,50,80)(30,101,41,71)>;
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), (1,52)(2,51)(3,60)(4,59)(5,58)(6,57)(7,56)(8,55)(9,54)(10,53)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,91)(20,100)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,50)(29,49)(30,48)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,70)(40,69)(71,103)(72,102)(73,101)(74,110)(75,109)(76,108)(77,107)(78,106)(79,105)(80,104)(81,119)(82,118)(83,117)(84,116)(85,115)(86,114)(87,113)(88,112)(89,111)(90,120), (1,65,29,53,35,50)(2,66,30,54,36,41)(3,67,21,55,37,42)(4,68,22,56,38,43)(5,69,23,57,39,44)(6,70,24,58,40,45)(7,61,25,59,31,46)(8,62,26,60,32,47)(9,63,27,51,33,48)(10,64,28,52,34,49)(11,83,110,95,113,80)(12,84,101,96,114,71)(13,85,102,97,115,72)(14,86,103,98,116,73)(15,87,104,99,117,74)(16,88,105,100,118,75)(17,89,106,91,119,76)(18,90,107,92,120,77)(19,81,108,93,111,78)(20,82,109,94,112,79), (1,113,53,83)(2,114,54,84)(3,115,55,85)(4,116,56,86)(5,117,57,87)(6,118,58,88)(7,119,59,89)(8,120,60,90)(9,111,51,81)(10,112,52,82)(11,65,95,35)(12,66,96,36)(13,67,97,37)(14,68,98,38)(15,69,99,39)(16,70,100,40)(17,61,91,31)(18,62,92,32)(19,63,93,33)(20,64,94,34)(21,102,42,72)(22,103,43,73)(23,104,44,74)(24,105,45,75)(25,106,46,76)(26,107,47,77)(27,108,48,78)(28,109,49,79)(29,110,50,80)(30,101,41,71) );
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)], [(1,52),(2,51),(3,60),(4,59),(5,58),(6,57),(7,56),(8,55),(9,54),(10,53),(11,99),(12,98),(13,97),(14,96),(15,95),(16,94),(17,93),(18,92),(19,91),(20,100),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,50),(29,49),(30,48),(31,68),(32,67),(33,66),(34,65),(35,64),(36,63),(37,62),(38,61),(39,70),(40,69),(71,103),(72,102),(73,101),(74,110),(75,109),(76,108),(77,107),(78,106),(79,105),(80,104),(81,119),(82,118),(83,117),(84,116),(85,115),(86,114),(87,113),(88,112),(89,111),(90,120)], [(1,65,29,53,35,50),(2,66,30,54,36,41),(3,67,21,55,37,42),(4,68,22,56,38,43),(5,69,23,57,39,44),(6,70,24,58,40,45),(7,61,25,59,31,46),(8,62,26,60,32,47),(9,63,27,51,33,48),(10,64,28,52,34,49),(11,83,110,95,113,80),(12,84,101,96,114,71),(13,85,102,97,115,72),(14,86,103,98,116,73),(15,87,104,99,117,74),(16,88,105,100,118,75),(17,89,106,91,119,76),(18,90,107,92,120,77),(19,81,108,93,111,78),(20,82,109,94,112,79)], [(1,113,53,83),(2,114,54,84),(3,115,55,85),(4,116,56,86),(5,117,57,87),(6,118,58,88),(7,119,59,89),(8,120,60,90),(9,111,51,81),(10,112,52,82),(11,65,95,35),(12,66,96,36),(13,67,97,37),(14,68,98,38),(15,69,99,39),(16,70,100,40),(17,61,91,31),(18,62,92,32),(19,63,93,33),(20,64,94,34),(21,102,42,72),(22,103,43,73),(23,104,44,74),(24,105,45,75),(25,106,46,76),(26,107,47,77),(27,108,48,78),(28,109,49,79),(29,110,50,80),(30,101,41,71)]])
D10⋊Dic3 is a maximal subgroup of
Dic3⋊C4⋊D5 D10⋊Dic6 Dic3.D20 D30.34D4 D30.D4 (D5×C12)⋊C4 (C4×D5)⋊Dic3 C60.67D4 C60.68D4 (C2×C12).D10 (C2×C60).C22 (C4×Dic3)⋊D5 C60.44D4 (C4×Dic15)⋊C2 C60.88D4 (D5×Dic3)⋊C4 D10.19(C4×S3) (C6×D5).D4 Dic15⋊D4 Dic3⋊D20 D10⋊1Dic6 D10⋊2Dic6 Dic3×D20 Dic15.D4 D10⋊4Dic6 D20⋊8Dic3 C4×C15⋊D4 D6⋊(C4×D5) C4×C3⋊D20 C15⋊20(C4×D4) D6⋊C4⋊D5 D10⋊C4⋊S3 (C2×Dic6)⋊D5 C60⋊4D4 D6.9D20 Dic15.10D4 Dic15.31D4 C12⋊2D20 S3×D10⋊C4 D30.27D4 D6⋊4D20 D30⋊6D4 C6.(D4×D5) (C2×C30).D4 C6.(C2×D20) D5×C6.D4 C23.17(S3×D5) Dic3×C5⋊D4 Dic15⋊16D4 (C2×C30)⋊D4 (C2×C6)⋊8D20 (S3×C10)⋊D4 (C2×C6)⋊D20 Dic15⋊18D4 D30⋊8D4
D10⋊Dic3 is a maximal quotient of
C60.93D4 C60.28D4 C12.6D20 C30.D8 C6.D40 C30.Q16 C6.Dic20 C60.96D4 C60.97D4 C30.24C42 (C2×C6).D20
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 15A | 15B | 20A | ··· | 20H | 30A | ··· | 30F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 6 | 6 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | Dic3 | D6 | D10 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | S3×D5 | D5×Dic3 | C15⋊D4 | C3⋊D20 |
| kernel | D10⋊Dic3 | C10×Dic3 | C2×Dic15 | D5×C2×C6 | C6×D5 | C22×D5 | C30 | C2×Dic3 | D10 | C2×C10 | C2×C6 | C10 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 |
Matrix representation of D10⋊Dic3 ►in GL4(𝔽61) generated by
| 17 | 17 | 0 | 0 |
| 44 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 |
| 44 | 44 | 0 | 0 |
| 60 | 17 | 0 | 0 |
| 0 | 0 | 60 | 0 |
| 0 | 0 | 23 | 1 |
| 60 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 |
| 0 | 0 | 13 | 0 |
| 0 | 0 | 25 | 47 |
| 29 | 54 | 0 | 0 |
| 7 | 32 | 0 | 0 |
| 0 | 0 | 27 | 5 |
| 0 | 0 | 13 | 34 |
G:=sub<GL(4,GF(61))| [17,44,0,0,17,1,0,0,0,0,60,0,0,0,0,60],[44,60,0,0,44,17,0,0,0,0,60,23,0,0,0,1],[60,0,0,0,0,60,0,0,0,0,13,25,0,0,0,47],[29,7,0,0,54,32,0,0,0,0,27,13,0,0,5,34] >;
D10⋊Dic3 in GAP, Magma, Sage, TeX
D_{10}\rtimes {\rm Dic}_3 % in TeX
G:=Group("D10:Dic3"); // GroupNames label
G:=SmallGroup(240,26);
// by ID
G=gap.SmallGroup(240,26);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-5,121,31,490,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^5*b,d*c*d^-1=c^-1>;
// generators/relations