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## G = Dic3≀C2order 288 = 25·32

### Wreath product of Dic3 by C2

Aliases: Dic3C2, C62.5D4, C322C4≀C2, Dic328C2, D6⋊S32C4, C22.3S3≀C2, C322Q82C4, C3⋊Dic3.6D4, C62.C43C2, D6.4D6.1C2, C2.11(S32⋊C4), C3⋊Dic3.11(C2×C4), (C3×C6).11(C22⋊C4), (C2×C3⋊Dic3).3C22, SmallGroup(288,389)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3⋊Dic3 — Dic3≀C2
 Chief series C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — D6.4D6 — Dic3≀C2
 Lower central C32 — C3×C6 — C3⋊Dic3 — Dic3≀C2
 Upper central C1 — C2 — C22

Generators and relations for Dic3≀C2
G = < a,b,c,d | a6=b6=d2=1, c4=b3, ab=ba, cac-1=a3b, dad=a-1b3, cbc-1=a2b3, bd=db, dcd=a3b3c3 >

Subgroups: 360 in 79 conjugacy classes, 15 normal (all characteristic)
C1, C2, C2 [×2], C3 [×2], C4 [×5], C22, C22, S3, C6 [×6], C8, C2×C4 [×3], D4 [×2], Q8, C32, Dic3 [×7], C12 [×3], D6, C2×C6 [×3], C42, M4(2), C4○D4, C3×S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3 [×4], C3⋊D4 [×2], C2×C12, C3×D4, C4≀C2, C3×Dic3 [×3], C3⋊Dic3 [×2], S3×C6, C62, C4×Dic3, D42S3, C322C8, S3×Dic3, D6⋊S3, C322Q8, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, Dic32, C62.C4, D6.4D6, Dic3≀C2
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, C4≀C2, S3≀C2, S32⋊C4, Dic3≀C2

Character table of Dic3≀C2

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 6D 6E 6F 8A 8B 12A 12B 12C 12D 12E size 1 1 2 12 4 4 6 6 6 6 9 9 12 18 4 4 4 4 8 24 36 36 12 12 12 12 24 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 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 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 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 linear of order 2 ρ5 1 1 -1 -1 1 1 i -i -i i -1 -1 1 1 1 -1 -1 1 -1 -1 i -i -i i -i i 1 linear of order 4 ρ6 1 1 -1 -1 1 1 -i i i -i -1 -1 1 1 1 -1 -1 1 -1 -1 -i i i -i i -i 1 linear of order 4 ρ7 1 1 -1 1 1 1 -i i i -i -1 -1 -1 1 1 -1 -1 1 -1 1 i -i i -i i -i -1 linear of order 4 ρ8 1 1 -1 1 1 1 i -i -i i -1 -1 -1 1 1 -1 -1 1 -1 1 -i i -i i -i i -1 linear of order 4 ρ9 2 2 -2 0 2 2 0 0 0 0 2 2 0 -2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 2 2 0 0 0 0 -2 -2 0 -2 2 2 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 -2 0 0 2 2 -1-i -1+i 1-i 1+i 2i -2i 0 0 -2 0 0 -2 0 0 0 0 -1+i 1+i 1-i -1-i 0 complex lifted from C4≀C2 ρ12 2 -2 0 0 2 2 1+i 1-i -1+i -1-i 2i -2i 0 0 -2 0 0 -2 0 0 0 0 1-i -1-i -1+i 1+i 0 complex lifted from C4≀C2 ρ13 2 -2 0 0 2 2 -1+i -1-i 1+i 1-i -2i 2i 0 0 -2 0 0 -2 0 0 0 0 -1-i 1-i 1+i -1+i 0 complex lifted from C4≀C2 ρ14 2 -2 0 0 2 2 1-i 1+i -1-i -1+i -2i 2i 0 0 -2 0 0 -2 0 0 0 0 1+i -1+i -1-i 1-i 0 complex lifted from C4≀C2 ρ15 4 4 -4 2 -2 1 0 0 0 0 0 0 -2 0 1 2 2 -2 -1 -1 0 0 0 0 0 0 1 orthogonal lifted from S32⋊C4 ρ16 4 4 4 0 1 -2 2 2 2 2 0 0 0 0 -2 1 1 1 -2 0 0 0 -1 -1 -1 -1 0 orthogonal lifted from S3≀C2 ρ17 4 4 4 2 -2 1 0 0 0 0 0 0 2 0 1 -2 -2 -2 1 -1 0 0 0 0 0 0 -1 orthogonal lifted from S3≀C2 ρ18 4 4 4 -2 -2 1 0 0 0 0 0 0 -2 0 1 -2 -2 -2 1 1 0 0 0 0 0 0 1 orthogonal lifted from S3≀C2 ρ19 4 4 4 0 1 -2 -2 -2 -2 -2 0 0 0 0 -2 1 1 1 -2 0 0 0 1 1 1 1 0 orthogonal lifted from S3≀C2 ρ20 4 4 -4 -2 -2 1 0 0 0 0 0 0 2 0 1 2 2 -2 -1 1 0 0 0 0 0 0 -1 orthogonal lifted from S32⋊C4 ρ21 4 -4 0 0 1 -2 -2 -2 2 2 0 0 0 0 2 -3 3 -1 0 0 0 0 1 -1 -1 1 0 symplectic faithful, Schur index 2 ρ22 4 -4 0 0 1 -2 2 2 -2 -2 0 0 0 0 2 -3 3 -1 0 0 0 0 -1 1 1 -1 0 symplectic faithful, Schur index 2 ρ23 4 -4 0 0 1 -2 2i -2i 2i -2i 0 0 0 0 2 3 -3 -1 0 0 0 0 i i -i -i 0 complex faithful ρ24 4 4 -4 0 1 -2 2i -2i -2i 2i 0 0 0 0 -2 -1 -1 1 2 0 0 0 i -i i -i 0 complex lifted from S32⋊C4 ρ25 4 4 -4 0 1 -2 -2i 2i 2i -2i 0 0 0 0 -2 -1 -1 1 2 0 0 0 -i i -i i 0 complex lifted from S32⋊C4 ρ26 4 -4 0 0 1 -2 -2i 2i -2i 2i 0 0 0 0 2 3 -3 -1 0 0 0 0 -i -i i i 0 complex faithful ρ27 8 -8 0 0 -4 2 0 0 0 0 0 0 0 0 -2 0 0 4 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Permutation representations of Dic3≀C2
On 24 points - transitive group 24T600
Generators in S24
(1 11 22)(2 16 23 6 12 19)(3 24 13)(4 21 14 8 17 10)(5 15 18)(7 20 9)
(1 15 22 5 11 18)(2 19 12 6 23 16)(3 20 13 7 24 9)(4 10 17 8 14 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 4)(2 7)(3 6)(5 8)(9 12)(10 15)(11 14)(13 16)(17 22)(18 21)(19 24)(20 23)

G:=sub<Sym(24)| (1,11,22)(2,16,23,6,12,19)(3,24,13)(4,21,14,8,17,10)(5,15,18)(7,20,9), (1,15,22,5,11,18)(2,19,12,6,23,16)(3,20,13,7,24,9)(4,10,17,8,14,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,4)(2,7)(3,6)(5,8)(9,12)(10,15)(11,14)(13,16)(17,22)(18,21)(19,24)(20,23)>;

G:=Group( (1,11,22)(2,16,23,6,12,19)(3,24,13)(4,21,14,8,17,10)(5,15,18)(7,20,9), (1,15,22,5,11,18)(2,19,12,6,23,16)(3,20,13,7,24,9)(4,10,17,8,14,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,4)(2,7)(3,6)(5,8)(9,12)(10,15)(11,14)(13,16)(17,22)(18,21)(19,24)(20,23) );

G=PermutationGroup([(1,11,22),(2,16,23,6,12,19),(3,24,13),(4,21,14,8,17,10),(5,15,18),(7,20,9)], [(1,15,22,5,11,18),(2,19,12,6,23,16),(3,20,13,7,24,9),(4,10,17,8,14,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,4),(2,7),(3,6),(5,8),(9,12),(10,15),(11,14),(13,16),(17,22),(18,21),(19,24),(20,23)])

G:=TransitiveGroup(24,600);

On 24 points - transitive group 24T604
Generators in S24
(1 11 22)(2 19 12 6 23 16)(3 24 13)(4 10 17 8 14 21)(5 15 18)(7 20 9)
(1 18 11 5 22 15)(2 19 12 6 23 16)(3 9 24 7 13 20)(4 10 17 8 14 21)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(1 4)(2 7)(3 6)(5 8)(9 23)(10 18)(11 17)(12 20)(13 19)(14 22)(15 21)(16 24)

G:=sub<Sym(24)| (1,11,22)(2,19,12,6,23,16)(3,24,13)(4,10,17,8,14,21)(5,15,18)(7,20,9), (1,18,11,5,22,15)(2,19,12,6,23,16)(3,9,24,7,13,20)(4,10,17,8,14,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,4)(2,7)(3,6)(5,8)(9,23)(10,18)(11,17)(12,20)(13,19)(14,22)(15,21)(16,24)>;

G:=Group( (1,11,22)(2,19,12,6,23,16)(3,24,13)(4,10,17,8,14,21)(5,15,18)(7,20,9), (1,18,11,5,22,15)(2,19,12,6,23,16)(3,9,24,7,13,20)(4,10,17,8,14,21), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (1,4)(2,7)(3,6)(5,8)(9,23)(10,18)(11,17)(12,20)(13,19)(14,22)(15,21)(16,24) );

G=PermutationGroup([(1,11,22),(2,19,12,6,23,16),(3,24,13),(4,10,17,8,14,21),(5,15,18),(7,20,9)], [(1,18,11,5,22,15),(2,19,12,6,23,16),(3,9,24,7,13,20),(4,10,17,8,14,21)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(1,4),(2,7),(3,6),(5,8),(9,23),(10,18),(11,17),(12,20),(13,19),(14,22),(15,21),(16,24)])

G:=TransitiveGroup(24,604);

Matrix representation of Dic3≀C2 in GL4(𝔽5) generated by

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

Dic3≀C2 in GAP, Magma, Sage, TeX

{\rm Dic}_3\wr C_2
% in TeX

G:=Group("Dic3wrC2");
// GroupNames label

G:=SmallGroup(288,389);
// by ID

G=gap.SmallGroup(288,389);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,85,64,422,100,675,2693,2028,691,797,2372]);
// Polycyclic

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

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