metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D20⋊5D6, Dic6⋊5D10, D30.39D4, C60.22C23, D60.6C22, Dic15.13D4, C3⋊C8⋊9D10, D4⋊D5⋊6S3, C5⋊2C8⋊9D6, (D4×D15)⋊2C2, C3⋊D40⋊4C2, C5⋊4(D8⋊S3), D4.S3⋊6D5, C6.72(D4×D5), C3⋊3(D40⋊C2), D4.16(S3×D5), (C5×D4).10D6, C10.73(S3×D4), D20⋊S3⋊2C2, C15⋊20(C8⋊C22), (C3×D4).10D10, C30.184(C2×D4), Dic6⋊D5⋊4C2, (C3×D20)⋊5C22, D30.5C4⋊3C2, C20.22(C22×S3), (C5×Dic6)⋊5C22, (C4×D15).6C22, C12.22(C22×D5), (D4×C15).16C22, C2.25(D10⋊D6), C4.22(C2×S3×D5), (C3×D4⋊D5)⋊8C2, (C5×C3⋊C8)⋊9C22, (C5×D4.S3)⋊8C2, (C3×C5⋊2C8)⋊9C22, SmallGroup(480,574)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6⋊D10
G = < a,b,c,d | a12=c10=d2=1, b2=a6, bab-1=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=a9b, dcd=c-1 >
Subgroups: 1004 in 136 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, M4(2), D8, SD16, C2×D4, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C3×D4, C3×D4, C22×S3, C3×D5, D15, C30, C30, C8⋊C22, C5⋊2C8, C40, C4×D5, D20, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C8⋊S3, C24⋊C2, D4⋊S3, D4.S3, C3×D8, S3×D4, D4⋊2S3, C5×Dic3, Dic15, C60, C6×D5, D30, D30, C2×C30, C8⋊D5, D40, D4⋊D5, Q8⋊D5, C5×SD16, D4×D5, Q8⋊2D5, D8⋊S3, C5×C3⋊C8, C3×C5⋊2C8, D5×Dic3, C3⋊D20, C3×D20, C5×Dic6, C4×D15, D60, C15⋊7D4, D4×C15, C22×D15, D40⋊C2, D30.5C4, C3⋊D40, Dic6⋊D5, C3×D4⋊D5, C5×D4.S3, D20⋊S3, D4×D15, Dic6⋊D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C8⋊C22, C22×D5, S3×D4, S3×D5, D4×D5, D8⋊S3, C2×S3×D5, D40⋊C2, D10⋊D6, Dic6⋊D10
►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 85 7 91)(2 96 8 90)(3 95 9 89)(4 94 10 88)(5 93 11 87)(6 92 12 86)(13 97 19 103)(14 108 20 102)(15 107 21 101)(16 106 22 100)(17 105 23 99)(18 104 24 98)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)(49 74 55 80)(50 73 56 79)(51 84 57 78)(52 83 58 77)(53 82 59 76)(54 81 60 75)(61 113 67 119)(62 112 68 118)(63 111 69 117)(64 110 70 116)(65 109 71 115)(66 120 72 114)
(1 83 63 42 14 4 80 66 39 17)(2 78 64 37 15 11 81 61 40 24)(3 73 65 44 16 6 82 68 41 19)(5 75 67 46 18 8 84 70 43 21)(7 77 69 48 20 10 74 72 45 23)(9 79 71 38 22 12 76 62 47 13)(25 104 87 57 113 31 98 93 51 119)(26 99 88 52 114)(27 106 89 59 115 33 100 95 53 109)(28 101 90 54 116)(29 108 91 49 117 35 102 85 55 111)(30 103 92 56 118)(32 105 94 58 120)(34 107 96 60 110)(36 97 86 50 112)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 60)(37 73)(38 84)(39 83)(40 82)(41 81)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(85 108)(86 107)(87 106)(88 105)(89 104)(90 103)(91 102)(92 101)(93 100)(94 99)(95 98)(96 97)(109 113)(110 112)(114 120)(115 119)(116 118)
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), (1,85,7,91)(2,96,8,90)(3,95,9,89)(4,94,10,88)(5,93,11,87)(6,92,12,86)(13,97,19,103)(14,108,20,102)(15,107,21,101)(16,106,22,100)(17,105,23,99)(18,104,24,98)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)(49,74,55,80)(50,73,56,79)(51,84,57,78)(52,83,58,77)(53,82,59,76)(54,81,60,75)(61,113,67,119)(62,112,68,118)(63,111,69,117)(64,110,70,116)(65,109,71,115)(66,120,72,114), (1,83,63,42,14,4,80,66,39,17)(2,78,64,37,15,11,81,61,40,24)(3,73,65,44,16,6,82,68,41,19)(5,75,67,46,18,8,84,70,43,21)(7,77,69,48,20,10,74,72,45,23)(9,79,71,38,22,12,76,62,47,13)(25,104,87,57,113,31,98,93,51,119)(26,99,88,52,114)(27,106,89,59,115,33,100,95,53,109)(28,101,90,54,116)(29,108,91,49,117,35,102,85,55,111)(30,103,92,56,118)(32,105,94,58,120)(34,107,96,60,110)(36,97,86,50,112), (1,17)(2,16)(3,15)(4,14)(5,13)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,60)(37,73)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(109,113)(110,112)(114,120)(115,119)(116,118)>;
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), (1,85,7,91)(2,96,8,90)(3,95,9,89)(4,94,10,88)(5,93,11,87)(6,92,12,86)(13,97,19,103)(14,108,20,102)(15,107,21,101)(16,106,22,100)(17,105,23,99)(18,104,24,98)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)(49,74,55,80)(50,73,56,79)(51,84,57,78)(52,83,58,77)(53,82,59,76)(54,81,60,75)(61,113,67,119)(62,112,68,118)(63,111,69,117)(64,110,70,116)(65,109,71,115)(66,120,72,114), (1,83,63,42,14,4,80,66,39,17)(2,78,64,37,15,11,81,61,40,24)(3,73,65,44,16,6,82,68,41,19)(5,75,67,46,18,8,84,70,43,21)(7,77,69,48,20,10,74,72,45,23)(9,79,71,38,22,12,76,62,47,13)(25,104,87,57,113,31,98,93,51,119)(26,99,88,52,114)(27,106,89,59,115,33,100,95,53,109)(28,101,90,54,116)(29,108,91,49,117,35,102,85,55,111)(30,103,92,56,118)(32,105,94,58,120)(34,107,96,60,110)(36,97,86,50,112), (1,17)(2,16)(3,15)(4,14)(5,13)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(25,59)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(32,52)(33,51)(34,50)(35,49)(36,60)(37,73)(38,84)(39,83)(40,82)(41,81)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(61,68)(62,67)(63,66)(64,65)(69,72)(70,71)(85,108)(86,107)(87,106)(88,105)(89,104)(90,103)(91,102)(92,101)(93,100)(94,99)(95,98)(96,97)(109,113)(110,112)(114,120)(115,119)(116,118) );
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)], [(1,85,7,91),(2,96,8,90),(3,95,9,89),(4,94,10,88),(5,93,11,87),(6,92,12,86),(13,97,19,103),(14,108,20,102),(15,107,21,101),(16,106,22,100),(17,105,23,99),(18,104,24,98),(25,43,31,37),(26,42,32,48),(27,41,33,47),(28,40,34,46),(29,39,35,45),(30,38,36,44),(49,74,55,80),(50,73,56,79),(51,84,57,78),(52,83,58,77),(53,82,59,76),(54,81,60,75),(61,113,67,119),(62,112,68,118),(63,111,69,117),(64,110,70,116),(65,109,71,115),(66,120,72,114)], [(1,83,63,42,14,4,80,66,39,17),(2,78,64,37,15,11,81,61,40,24),(3,73,65,44,16,6,82,68,41,19),(5,75,67,46,18,8,84,70,43,21),(7,77,69,48,20,10,74,72,45,23),(9,79,71,38,22,12,76,62,47,13),(25,104,87,57,113,31,98,93,51,119),(26,99,88,52,114),(27,106,89,59,115,33,100,95,53,109),(28,101,90,54,116),(29,108,91,49,117,35,102,85,55,111),(30,103,92,56,118),(32,105,94,58,120),(34,107,96,60,110),(36,97,86,50,112)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(25,59),(26,58),(27,57),(28,56),(29,55),(30,54),(31,53),(32,52),(33,51),(34,50),(35,49),(36,60),(37,73),(38,84),(39,83),(40,82),(41,81),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(61,68),(62,67),(63,66),(64,65),(69,72),(70,71),(85,108),(86,107),(87,106),(88,105),(89,104),(90,103),(91,102),(92,101),(93,100),(94,99),(95,98),(96,97),(109,113),(110,112),(114,120),(115,119),(116,118)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12 | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 20 | 30 | 60 | 2 | 2 | 12 | 30 | 2 | 2 | 2 | 8 | 40 | 12 | 20 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 4 | 24 | 24 | 20 | 20 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C8⋊C22 | S3×D4 | S3×D5 | D4×D5 | D8⋊S3 | C2×S3×D5 | D40⋊C2 | D10⋊D6 | Dic6⋊D10 |
| kernel | Dic6⋊D10 | D30.5C4 | C3⋊D40 | Dic6⋊D5 | C3×D4⋊D5 | C5×D4.S3 | D20⋊S3 | D4×D15 | D4⋊D5 | Dic15 | D30 | D4.S3 | C5⋊2C8 | D20 | C5×D4 | C3⋊C8 | Dic6 | C3×D4 | C15 | C10 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of Dic6⋊D10 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 94 | 194 | 147 |
| 0 | 0 | 47 | 194 | 194 | 47 |
| 0 | 0 | 194 | 147 | 194 | 147 |
| 0 | 0 | 194 | 47 | 194 | 47 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 240 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,240,1,0,0,0,240,0,0,0,0,1,240,0,0],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,47,47,194,194,0,0,94,194,147,47,0,0,194,194,194,194,0,0,147,47,147,47],[190,51,0,0,0,0,190,240,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[190,240,0,0,0,0,190,51,0,0,0,0,0,0,0,0,1,240,0,0,0,0,0,240,0,0,1,240,0,0,0,0,0,240,0,0] >;
Dic6⋊D10 in GAP, Magma, Sage, TeX
{\rm Dic}_6\rtimes D_{10} % in TeX
G:=Group("Dic6:D10"); // GroupNames label
G:=SmallGroup(480,574);
// by ID
G=gap.SmallGroup(480,574);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,303,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=c^10=d^2=1,b^2=a^6,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a^9*b,d*c*d=c^-1>;
// generators/relations