direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×Dic3, C23.22D6, C3⋊5(C4×D4), C12⋊3(C2×C4), (C3×D4)⋊3C4, C2.5(S3×D4), (C6×D4).4C2, (C2×D4).7S3, C4⋊1(C2×Dic3), (C2×C4).49D6, C6.37(C2×D4), C4⋊Dic3⋊13C2, (C4×Dic3)⋊4C2, C6.28(C4○D4), C6.D4⋊7C2, (C2×C6).49C23, C6.25(C22×C4), C22⋊2(C2×Dic3), C2.5(D4⋊2S3), (C2×C12).32C22, (C22×Dic3)⋊4C2, C2.6(C22×Dic3), C22.25(C22×S3), (C22×C6).17C22, (C2×Dic3).37C22, (C2×C6)⋊3(C2×C4), (C2×D4)○(C2×Dic3), SmallGroup(96,141)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×Dic3
G = < a,b,c,d | a4=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 178 in 94 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, D4, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Dic3, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C4×D4, C4×Dic3, C4⋊Dic3, C6.D4, C22×Dic3, C6×D4, D4×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22×C4, C2×D4, C4○D4, C2×Dic3, C22×S3, C4×D4, S3×D4, D4⋊2S3, C22×Dic3, D4×Dic3
Character table of D4×Dic3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | -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 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -i | i | -i | i | -i | -i | i | -i | i | i | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ10 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -i | i | -i | i | -i | i | -i | i | i | -i | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | i | -i | i | -i | i | i | -i | i | -i | -i | -1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ12 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | i | -i | i | -i | i | -i | i | -i | -i | i | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | i | -i | i | -i | -i | i | i | -i | i | -i | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | i | -i | -i | -i | -i | i | i | i | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | -i | i | -i | i | i | -i | -i | i | -i | i | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | -i | i | i | i | i | -i | -i | -i | -1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ17 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | 2 | 2 | -2 | 2 | 2 | -2 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ22 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ23 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | -2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ24 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ25 | 2 | 2 | -2 | -2 | -2 | 2 | -2 | 2 | -1 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | symplectic lifted from Dic3, Schur index 2 |
| ρ26 | 2 | 2 | -2 | -2 | -2 | -2 | 2 | 2 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ27 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ28 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 2i | -2i | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ30 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
►On 48 points
(1 27 16 24)(2 28 17 19)(3 29 18 20)(4 30 13 21)(5 25 14 22)(6 26 15 23)(7 38 45 33)(8 39 46 34)(9 40 47 35)(10 41 48 36)(11 42 43 31)(12 37 44 32)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 33 4 36)(2 32 5 35)(3 31 6 34)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)
G:=sub<Sym(48)| (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,38,45,33)(8,39,46,34)(9,40,47,35)(10,41,48,36)(11,42,43,31)(12,37,44,32), (1,24)(2,19)(3,20)(4,21)(5,22)(6,23)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45)>;
G:=Group( (1,27,16,24)(2,28,17,19)(3,29,18,20)(4,30,13,21)(5,25,14,22)(6,26,15,23)(7,38,45,33)(8,39,46,34)(9,40,47,35)(10,41,48,36)(11,42,43,31)(12,37,44,32), (1,24)(2,19)(3,20)(4,21)(5,22)(6,23)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,33,4,36)(2,32,5,35)(3,31,6,34)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45) );
G=PermutationGroup([[(1,27,16,24),(2,28,17,19),(3,29,18,20),(4,30,13,21),(5,25,14,22),(6,26,15,23),(7,38,45,33),(8,39,46,34),(9,40,47,35),(10,41,48,36),(11,42,43,31),(12,37,44,32)], [(1,24),(2,19),(3,20),(4,21),(5,22),(6,23),(7,38),(8,39),(9,40),(10,41),(11,42),(12,37),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,33,4,36),(2,32,5,35),(3,31,6,34),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45)]])
D4×Dic3 is a maximal subgroup of
Dic3⋊4D8 D4.S3⋊C4 Dic3⋊6SD16 Dic3.D8 D4⋊Dic6 D4.Dic6 D4.2Dic6 D4⋊S3⋊C4 Dic3⋊D8 D8⋊Dic3 (C6×D8).C2 Dic3⋊3SD16 SD16⋊Dic3 (C3×D4).D4 C4×D4⋊2S3 D4⋊5Dic6 D4⋊6Dic6 C4×S3×D4 C42⋊13D6 C42.108D6 C24.67D6 C24.43D6 C24.44D6 C24.46D6 C12⋊(C4○D4) Dic6⋊19D4 C4⋊C4.178D6 C6.342+ 1+4 C6.702- 1+4 C6.712- 1+4 C4⋊C4⋊21D6 C6.732- 1+4 C6.432+ 1+4 C6.452+ 1+4 C6.462+ 1+4 C6.1152+ 1+4 C6.472+ 1+4 C4⋊C4.197D6 C6.802- 1+4 C6.1222+ 1+4 C6.852- 1+4 C42.139D6 C42.234D6 C42.143D6 C42.144D6 C42.166D6 C42.238D6 Dic6⋊11D4 C42.168D6 C24.49D6 C24.53D6 C6.1042- 1+4 C6.1442+ 1+4 C6.1452+ 1+4 D12⋊Dic3 C62.115C23 D20⋊8Dic3 Dic15⋊16D4
D4×Dic3 is a maximal quotient of
C24.58D6 C24.19D6 C4⋊C4⋊5Dic3 C4⋊C4⋊6Dic3 C42.47D6 C12⋊3M4(2) D8⋊Dic3 SD16⋊Dic3 Q16⋊Dic3 D8⋊5Dic3 D8⋊4Dic3 C24.29D6 C24.30D6 D12⋊Dic3 C62.115C23 D20⋊8Dic3 Dic15⋊16D4
Matrix representation of D4×Dic3 ►in GL5(𝔽13)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 5 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 | 12 |
G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,12,0,0,0,1,0],[5,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,1,1,0,0,0,0,12] >;
D4×Dic3 in GAP, Magma, Sage, TeX
D_4\times {\rm Dic}_3 % in TeX
G:=Group("D4xDic3"); // GroupNames label
G:=SmallGroup(96,141);
// by ID
G=gap.SmallGroup(96,141);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,188,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
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