metabelian, supersoluble, monomial
Aliases: D6⋊Dic3, C6.16D12, C62.2C22, (S3×C6)⋊1C4, C22.4S32, (C2×C6).9D6, C3⋊3(D6⋊C4), C6.19(C4×S3), (C3×C6).13D4, (C22×S3).S3, (C2×Dic3)⋊1S3, (C6×Dic3)⋊1C2, C6.4(C2×Dic3), C2.4(S3×Dic3), C6.10(C3⋊D4), C32⋊3(C22⋊C4), C2.1(D6⋊S3), C2.1(C3⋊D12), C3⋊1(C6.D4), (S3×C2×C6).1C2, (C3×C6).13(C2×C4), (C2×C3⋊Dic3)⋊1C2, SmallGroup(144,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Dic3
G = < a,b,c,d | a6=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 224 in 76 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×C6, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C62, D6⋊C4, C6.D4, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, D6⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, D6⋊Dic3
Character table of D6⋊Dic3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 4 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ8 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S3 |
| ρ12 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | -2 | 2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | 1 | 1 | -2 | -2 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | 2 | -2 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | 1 | 1 | -1 | √-3 | √-3 | -√-3 | -√-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | -1 | 1 | 1 | -√-3 | √-3 | -√-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | -√-3 | √-3 | complex lifted from C3⋊D4 |
| ρ22 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | -2 | 1 | -2 | 2 | -1 | 1 | -1 | 1 | 1 | √-3 | -√-3 | √-3 | -√-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ23 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -2i | 2i | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ24 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | 2i | -2i | 0 | 0 | 1 | -2 | -1 | 1 | -2 | 2 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ25 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 2 | -1 | -2 | -2 | 1 | 1 | 1 | 1 | -1 | -√-3 | -√-3 | √-3 | √-3 | 0 | 0 | 0 | 0 | complex lifted from C3⋊D4 |
| ρ26 | 2 | 2 | -2 | -2 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 2 | -2 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | √-3 | -√-3 | complex lifted from C3⋊D4 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 2 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C3⋊D12 |
| ρ28 | 4 | 4 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ29 | 4 | 4 | -4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 2 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D6⋊S3, Schur index 2 |
| ρ30 | 4 | -4 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | 2 | -2 | -1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3×Dic3, Schur index 2 |
►On 48 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)
(1 34)(2 33)(3 32)(4 31)(5 36)(6 35)(7 28)(8 27)(9 26)(10 25)(11 30)(12 29)(13 46)(14 45)(15 44)(16 43)(17 48)(18 47)(19 40)(20 39)(21 38)(22 37)(23 42)(24 41)
(1 11 3 7 5 9)(2 12 4 8 6 10)(13 21 17 19 15 23)(14 22 18 20 16 24)(25 33 29 31 27 35)(26 34 30 32 28 36)(37 47 39 43 41 45)(38 48 40 44 42 46)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
G:=sub<Sym(48)| (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), (1,34)(2,33)(3,32)(4,31)(5,36)(6,35)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,21,17,19,15,23)(14,22,18,20,16,24)(25,33,29,31,27,35)(26,34,30,32,28,36)(37,47,39,43,41,45)(38,48,40,44,42,46), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
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), (1,34)(2,33)(3,32)(4,31)(5,36)(6,35)(7,28)(8,27)(9,26)(10,25)(11,30)(12,29)(13,46)(14,45)(15,44)(16,43)(17,48)(18,47)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41), (1,11,3,7,5,9)(2,12,4,8,6,10)(13,21,17,19,15,23)(14,22,18,20,16,24)(25,33,29,31,27,35)(26,34,30,32,28,36)(37,47,39,43,41,45)(38,48,40,44,42,46), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
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)], [(1,34),(2,33),(3,32),(4,31),(5,36),(6,35),(7,28),(8,27),(9,26),(10,25),(11,30),(12,29),(13,46),(14,45),(15,44),(16,43),(17,48),(18,47),(19,40),(20,39),(21,38),(22,37),(23,42),(24,41)], [(1,11,3,7,5,9),(2,12,4,8,6,10),(13,21,17,19,15,23),(14,22,18,20,16,24),(25,33,29,31,27,35),(26,34,30,32,28,36),(37,47,39,43,41,45),(38,48,40,44,42,46)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
D6⋊Dic3 is a maximal subgroup of
C62.11C23 C62.20C23 D6⋊Dic6 Dic3.D12 C62.23C23 C62.24C23 C62.25C23 D6⋊6Dic6 D6⋊7Dic6 C62.28C23 C62.29C23 C12.27D12 C62.31C23 C62.32C23 C62.33C23 C62.47C23 C62.48C23 C62.49C23 C62.54C23 C62.55C23 Dic3⋊D12 D6⋊1Dic6 D6.9D12 Dic3×D12 D6⋊2Dic6 D6⋊3Dic6 D12⋊Dic3 D6⋊4Dic6 C4×D6⋊S3 C4×C3⋊D12 C62.74C23 C62.75C23 C62.77C23 D6⋊2D12 C62.83C23 C62.85C23 C12⋊2D12 S3×D6⋊C4 C62.91C23 D6⋊4D12 C62.100C23 C62.101C23 C62.56D4 C62.57D4 S3×C6.D4 C62.111C23 C62.112C23 Dic3×C3⋊D4 C62.115C23 C62⋊4D4 C62⋊5D4 C62⋊6D4 C62⋊7D4 C62.125C23 D18⋊Dic3 D6⋊Dic9 C62.4D6 C62.5D6 C62.77D6 C62.78D6 C62.84D6
D6⋊Dic3 is a maximal quotient of
C12.77D12 C12.D12 C12.14D12 D12⋊3Dic3 C6.16D24 Dic6⋊Dic3 C6.Dic12 D12⋊4Dic3 D12⋊2Dic3 C62.6Q8 C62.31D4 D18⋊Dic3 D6⋊Dic9 C62.4D6 C62.77D6 C62.78D6 C62.84D6
Matrix representation of D6⋊Dic3 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D6⋊Dic3 in GAP, Magma, Sage, TeX
D_6\rtimes {\rm Dic}_3 % in TeX
G:=Group("D6:Dic3"); // GroupNames label
G:=SmallGroup(144,64);
// by ID
G=gap.SmallGroup(144,64);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^6=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^3*b,d*c*d^-1=c^-1>;
// generators/relations
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