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G = Dic3≀C2order 288 = 25·32

Wreath product of Dic3 by C2

non-abelian, soluble, monomial

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

C1C32C3⋊Dic3 — Dic3≀C2
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3D6.4D6 — Dic3≀C2
C32C3×C6C3⋊Dic3 — Dic3≀C2
C1C2C22

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 12A2B2C3A3B4A4B4C4D4E4F4G4H6A6B6C6D6E6F8A8B12A12B12C12D12E
 size 11212446666991218444482436361212121224
ρ1111111111111111111111111111    trivial
ρ2111111-1-1-1-11111111111-1-1-1-1-1-11    linear of order 2
ρ3111-111111111-1111111-1-1-11111-1    linear of order 2
ρ4111-111-1-1-1-111-1111111-111-1-1-1-1-1    linear of order 2
ρ511-1-111i-i-ii-1-1111-1-11-1-1i-i-ii-ii1    linear of order 4
ρ611-1-111-iii-i-1-1111-1-11-1-1-iii-ii-i1    linear of order 4
ρ711-1111-iii-i-1-1-111-1-11-11i-ii-ii-i-1    linear of order 4
ρ811-1111i-i-ii-1-1-111-1-11-11-ii-ii-ii-1    linear of order 4
ρ922-20220000220-22-2-22-200000000    orthogonal lifted from D4
ρ102220220000-2-20-22222200000000    orthogonal lifted from D4
ρ112-20022-1-i-1+i1-i1+i2i-2i00-200-20000-1+i1+i1-i-1-i0    complex lifted from C4≀C2
ρ122-200221+i1-i-1+i-1-i2i-2i00-200-200001-i-1-i-1+i1+i0    complex lifted from C4≀C2
ρ132-20022-1+i-1-i1+i1-i-2i2i00-200-20000-1-i1-i1+i-1+i0    complex lifted from C4≀C2
ρ142-200221-i1+i-1-i-1+i-2i2i00-200-200001+i-1+i-1-i1-i0    complex lifted from C4≀C2
ρ1544-42-21000000-20122-2-1-10000001    orthogonal lifted from S32⋊C4
ρ1644401-222220000-2111-2000-1-1-1-10    orthogonal lifted from S3≀C2
ρ174442-21000000201-2-2-21-1000000-1    orthogonal lifted from S3≀C2
ρ18444-2-21000000-201-2-2-2110000001    orthogonal lifted from S3≀C2
ρ1944401-2-2-2-2-20000-2111-200011110    orthogonal lifted from S3≀C2
ρ2044-4-2-2100000020122-2-11000000-1    orthogonal lifted from S32⋊C4
ρ214-4001-2-2-22200002-33-100001-1-110    symplectic faithful, Schur index 2
ρ224-4001-222-2-200002-33-10000-111-10    symplectic faithful, Schur index 2
ρ234-4001-22i-2i2i-2i000023-3-10000ii-i-i0    complex faithful
ρ2444-401-22i-2i-2i2i0000-2-1-112000i-ii-i0    complex lifted from S32⋊C4
ρ2544-401-2-2i2i2i-2i0000-2-1-112000-ii-ii0    complex lifted from S32⋊C4
ρ264-4001-2-2i2i-2i2i000023-3-10000-i-iii0    complex faithful
ρ278-800-4200000000-2004000000000    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

0004
0130
0300
1004
,
1004
0130
0300
1000
,
0100
2003
1000
0130
,
0030
4001
2000
0130
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

Export

Character table of Dic3≀C2 in TeX

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