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## G = Dic3⋊C4order 48 = 24·3

### The semidirect product of Dic3 and C4 acting via C4/C2=C2

Aliases: Dic3⋊C4, C6.5D4, C6.1Q8, C2.1Dic6, C22.4D6, C31(C4⋊C4), C2.4(C4×S3), C6.3(C2×C4), (C2×C4).1S3, (C2×C12).1C2, C2.1(C3⋊D4), (C2×C6).4C22, (C2×Dic3).1C2, SmallGroup(48,12)

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3⋊C4
G = < a,b,c | a6=c4=1, b2=a3, bab-1=a-1, ac=ca, cbc-1=a3b >

Character table of Dic3⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 2 2 2 6 6 6 6 2 2 2 2 2 2 2 ρ1 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 linear of order 2 ρ3 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 linear of order 2 ρ5 1 -1 1 -1 1 i -i -i -1 i 1 1 -1 -1 -i -i i i linear of order 4 ρ6 1 -1 1 -1 1 i -i i 1 -i -1 1 -1 -1 -i -i i i linear of order 4 ρ7 1 -1 1 -1 1 -i i -i 1 i -1 1 -1 -1 i i -i -i linear of order 4 ρ8 1 -1 1 -1 1 -i i i -1 -i 1 1 -1 -1 i i -i -i linear of order 4 ρ9 2 2 2 2 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 -2 -2 2 2 0 0 0 0 0 0 -2 2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ13 2 -2 -2 2 -1 0 0 0 0 0 0 1 -1 1 √3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ14 2 -2 -2 2 -1 0 0 0 0 0 0 1 -1 1 -√3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2 ρ15 2 -2 2 -2 -1 2i -2i 0 0 0 0 -1 1 1 i i -i -i complex lifted from C4×S3 ρ16 2 -2 2 -2 -1 -2i 2i 0 0 0 0 -1 1 1 -i -i i i complex lifted from C4×S3 ρ17 2 2 -2 -2 -1 0 0 0 0 0 0 1 1 -1 √-3 -√-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 -2 -2 -1 0 0 0 0 0 0 1 1 -1 -√-3 √-3 -√-3 √-3 complex lifted from C3⋊D4

Smallest permutation representation of Dic3⋊C4
Regular action 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 34 4 31)(2 33 5 36)(3 32 6 35)(7 26 10 29)(8 25 11 28)(9 30 12 27)(13 44 16 47)(14 43 17 46)(15 48 18 45)(19 38 22 41)(20 37 23 40)(21 42 24 39)
(1 23 11 17)(2 24 12 18)(3 19 7 13)(4 20 8 14)(5 21 9 15)(6 22 10 16)(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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(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), (1,34,4,31)(2,33,5,36)(3,32,6,35)(7,26,10,29)(8,25,11,28)(9,30,12,27)(13,44,16,47)(14,43,17,46)(15,48,18,45)(19,38,22,41)(20,37,23,40)(21,42,24,39), (1,23,11,17)(2,24,12,18)(3,19,7,13)(4,20,8,14)(5,21,9,15)(6,22,10,16)(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)], [(1,34,4,31),(2,33,5,36),(3,32,6,35),(7,26,10,29),(8,25,11,28),(9,30,12,27),(13,44,16,47),(14,43,17,46),(15,48,18,45),(19,38,22,41),(20,37,23,40),(21,42,24,39)], [(1,23,11,17),(2,24,12,18),(3,19,7,13),(4,20,8,14),(5,21,9,15),(6,22,10,16),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)])

Matrix representation of Dic3⋊C4 in GL3(𝔽13) generated by

 1 0 0 0 0 12 0 1 1
,
 12 0 0 0 9 11 0 2 4
,
 5 0 0 0 2 4 0 9 11
G:=sub<GL(3,GF(13))| [1,0,0,0,0,1,0,12,1],[12,0,0,0,9,2,0,11,4],[5,0,0,0,2,9,0,4,11] >;

Dic3⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes C_4
% in TeX

G:=Group("Dic3:C4");
// GroupNames label

G:=SmallGroup(48,12);
// by ID

G=gap.SmallGroup(48,12);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,40,101,26,804]);
// Polycyclic

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

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