Copied to
clipboard

## G = Dic3.D4order 96 = 25·3

### 1st non-split extension by Dic3 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — Dic3.D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — Dic3.D4
 Lower central C3 — C2×C6 — Dic3.D4
 Upper central C1 — C22 — C22⋊C4

Generators and relations for Dic3.D4
G = < a,b,c,d | a6=c4=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=a3c-1 >

Subgroups: 154 in 74 conjugacy classes, 35 normal (29 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×7], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, Dic3 [×2], Dic3 [×3], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×3], C22×C4, C2×Q8, Dic6 [×2], C2×Dic3 [×4], C2×Dic3 [×2], C2×C12 [×2], C22×C6, C22⋊Q8, Dic3⋊C4 [×2], C4⋊Dic3, C6.D4, C3×C22⋊C4, C2×Dic6, C22×Dic3, Dic3.D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, Dic6 [×2], C22×S3, C22⋊Q8, C2×Dic6, S3×D4, D42S3, Dic3.D4

Character table of Dic3.D4

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 12A 12B 12C 12D size 1 1 1 1 2 2 2 4 4 6 6 6 6 12 12 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 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 2 2 -1 -2 -2 0 0 0 0 0 0 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ10 2 -2 2 -2 0 0 2 0 0 0 2 0 -2 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -2 -2 -1 2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 2 2 -1 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ13 2 -2 2 -2 0 0 2 0 0 0 -2 0 2 0 0 2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 -2 -1 -2 2 0 0 0 0 0 0 -1 -1 -1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ15 2 2 -2 -2 2 -2 2 0 0 0 0 0 0 0 0 -2 2 -2 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ16 2 2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 -2 2 -2 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ17 2 2 -2 -2 -2 2 -1 0 0 0 0 0 0 0 0 1 -1 1 -1 1 -√3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ18 2 2 -2 -2 2 -2 -1 0 0 0 0 0 0 0 0 1 -1 1 1 -1 -√3 √3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ19 2 2 -2 -2 2 -2 -1 0 0 0 0 0 0 0 0 1 -1 1 1 -1 √3 -√3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ20 2 2 -2 -2 -2 2 -1 0 0 0 0 0 0 0 0 1 -1 1 -1 1 √3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ21 2 -2 -2 2 0 0 2 0 0 2i 0 -2i 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 0 0 2 0 0 -2i 0 2i 0 0 0 -2 -2 2 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 4 -4 0 0 -2 0 0 0 0 0 0 0 0 -2 2 2 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ24 4 -4 -4 4 0 0 -2 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

Smallest permutation representation of Dic3.D4
On 48 points
Generators in S48
(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 38 4 41)(2 37 5 40)(3 42 6 39)(7 21 10 24)(8 20 11 23)(9 19 12 22)(13 36 16 33)(14 35 17 32)(15 34 18 31)(25 47 28 44)(26 46 29 43)(27 45 30 48)
(1 21 13 27)(2 22 14 28)(3 23 15 29)(4 24 16 30)(5 19 17 25)(6 20 18 26)(7 36 48 38)(8 31 43 39)(9 32 44 40)(10 33 45 41)(11 34 46 42)(12 35 47 37)
(7 45)(8 46)(9 47)(10 48)(11 43)(12 44)(19 28)(20 29)(21 30)(22 25)(23 26)(24 27)

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), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48), (1,21,13,27)(2,22,14,28)(3,23,15,29)(4,24,16,30)(5,19,17,25)(6,20,18,26)(7,36,48,38)(8,31,43,39)(9,32,44,40)(10,33,45,41)(11,34,46,42)(12,35,47,37), (7,45)(8,46)(9,47)(10,48)(11,43)(12,44)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27)>;

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), (1,38,4,41)(2,37,5,40)(3,42,6,39)(7,21,10,24)(8,20,11,23)(9,19,12,22)(13,36,16,33)(14,35,17,32)(15,34,18,31)(25,47,28,44)(26,46,29,43)(27,45,30,48), (1,21,13,27)(2,22,14,28)(3,23,15,29)(4,24,16,30)(5,19,17,25)(6,20,18,26)(7,36,48,38)(8,31,43,39)(9,32,44,40)(10,33,45,41)(11,34,46,42)(12,35,47,37), (7,45)(8,46)(9,47)(10,48)(11,43)(12,44)(19,28)(20,29)(21,30)(22,25)(23,26)(24,27) );

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)], [(1,38,4,41),(2,37,5,40),(3,42,6,39),(7,21,10,24),(8,20,11,23),(9,19,12,22),(13,36,16,33),(14,35,17,32),(15,34,18,31),(25,47,28,44),(26,46,29,43),(27,45,30,48)], [(1,21,13,27),(2,22,14,28),(3,23,15,29),(4,24,16,30),(5,19,17,25),(6,20,18,26),(7,36,48,38),(8,31,43,39),(9,32,44,40),(10,33,45,41),(11,34,46,42),(12,35,47,37)], [(7,45),(8,46),(9,47),(10,48),(11,43),(12,44),(19,28),(20,29),(21,30),(22,25),(23,26),(24,27)])

Matrix representation of Dic3.D4 in GL4(𝔽13) generated by

 1 0 0 0 0 1 0 0 0 0 1 12 0 0 1 0
,
 12 0 0 0 0 12 0 0 0 0 0 5 0 0 5 0
,
 0 12 0 0 1 0 0 0 0 0 3 7 0 0 6 10
,
 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(13))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,12,0],[12,0,0,0,0,12,0,0,0,0,0,5,0,0,5,0],[0,1,0,0,12,0,0,0,0,0,3,6,0,0,7,10],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1] >;

Dic3.D4 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_4
% in TeX

G:=Group("Dic3.D4");
// GroupNames label

G:=SmallGroup(96,85);
// by ID

G=gap.SmallGroup(96,85);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,218,188,50,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^4=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=a^3*c^-1>;
// generators/relations

Export

׿
×
𝔽