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G = Dic3.D6order 144 = 24·32

2nd non-split extension by Dic3 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: Dic64S3, C12.23D6, Dic3.2D6, C4.12S32, C3⋊S32Q8, C31(S3×Q8), C323(C2×Q8), (C3×Dic6)⋊6C2, C322Q83C2, (C3×C6).4C23, C6.4(C22×S3), C6.D6.1C2, (C3×C12).19C22, C3⋊Dic3.12C22, (C3×Dic3).3C22, C2.7(C2×S32), (C4×C3⋊S3).1C2, (C2×C3⋊S3).12C22, SmallGroup(144,140)

Series: Derived Chief Lower central Upper central

C1C3×C6 — Dic3.D6
C1C3C32C3×C6C3×Dic3C6.D6 — Dic3.D6
C32C3×C6 — Dic3.D6
C1C2C4

Generators and relations for Dic3.D6
 G = < a,b,c,d | a6=1, b2=c6=d2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c5 >

Subgroups: 256 in 84 conjugacy classes, 34 normal (10 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4 [×5], C22, S3 [×6], C6 [×2], C6, C2×C4 [×3], Q8 [×4], C32, Dic3 [×4], Dic3 [×3], C12 [×2], C12 [×5], D6 [×3], C2×Q8, C3⋊S3 [×2], C3×C6, Dic6 [×2], Dic6 [×4], C4×S3 [×7], C3×Q8 [×2], C3×Dic3 [×4], C3⋊Dic3, C3×C12, C2×C3⋊S3, S3×Q8 [×2], C6.D6 [×2], C322Q8 [×2], C3×Dic6 [×2], C4×C3⋊S3, Dic3.D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, C22×S3 [×2], S32, S3×Q8 [×2], C2×S32, Dic3.D6

Character table of Dic3.D6

 class 12A2B2C3A3B3C4A4B4C4D4E4F6A6B6C12A12B12C12D12E12F12G12H
 size 11992242666618224444412121212
ρ1111111111111111111111111    trivial
ρ21111111-11-1-11-1111-1-1-1-1-111-1    linear of order 2
ρ31111111-1-111-1-1111-1-1-1-11-1-11    linear of order 2
ρ411111111-1-1-1-111111111-1-1-1-1    linear of order 2
ρ511-1-1111111-1-1-111111111-11-1    linear of order 2
ρ611-1-1111-11-11-11111-1-1-1-1-1-111    linear of order 2
ρ711-1-1111-1-11-111111-1-1-1-111-1-1    linear of order 2
ρ811-1-11111-1-111-11111111-11-11    linear of order 2
ρ922002-1-1-2-22000-12-1111-2-1010    orthogonal lifted from D6
ρ102200-12-12002202-1-1-12-1-10-10-1    orthogonal lifted from S3
ρ1122002-1-1-22-2000-12-1111-210-10    orthogonal lifted from D6
ρ1222002-1-12-2-2000-12-1-1-1-121010    orthogonal lifted from D6
ρ132200-12-1-200-2202-1-11-2110-101    orthogonal lifted from D6
ρ142200-12-1200-2-202-1-1-12-1-10101    orthogonal lifted from D6
ρ152200-12-1-2002-202-1-11-211010-1    orthogonal lifted from D6
ρ1622002-1-1222000-12-1-1-1-12-10-10    orthogonal lifted from S3
ρ172-22-2222000000-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ182-2-22222000000-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ194400-2-21-400000-2-21-12-120000    orthogonal lifted from C2×S32
ρ204400-2-21400000-2-211-21-20000    orthogonal lifted from S32
ρ214-400-24-2000000-42200000000    symplectic lifted from S3×Q8, Schur index 2
ρ224-4004-2-20000002-4200000000    symplectic lifted from S3×Q8, Schur index 2
ρ234-400-2-2100000022-13i0-3i00000    complex faithful
ρ244-400-2-2100000022-1-3i03i00000    complex faithful

Permutation representations of Dic3.D6
On 24 points - transitive group 24T223
Generators in S24
(1 11 9 7 5 3)(2 12 10 8 6 4)(13 15 17 19 21 23)(14 16 18 20 22 24)
(1 19 7 13)(2 14 8 20)(3 21 9 15)(4 16 10 22)(5 23 11 17)(6 18 12 24)
(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 18 7 24)(2 17 8 23)(3 16 9 22)(4 15 10 21)(5 14 11 20)(6 13 12 19)

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

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

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

G:=TransitiveGroup(24,223);

Dic3.D6 is a maximal subgroup of
(C3×C12).D4  C3⋊S3.2Q16  C249D6  C24.23D6  D12.4D6  Dic6⋊D6  Dic6.D6  Dic6.9D6  Dic6.10D6  S32⋊Q8  C4.4S3≀C2  C32⋊C4⋊Q8  C32⋊Q16⋊C2  C3⋊S32SD16  C3⋊S3⋊Q16  D12.33D6  D1223D6  Dic6.24D6  Dic612D6  Dic6.26D6  S32×Q8  Dic18⋊S3  C3⋊S3⋊Dic6  C335(C2×Q8)  C336(C2×Q8)  C329(S3×Q8)  C3⋊S34Dic6
Dic3.D6 is a maximal quotient of
C62.8C23  C62.9C23  C62.13C23  C62.17C23  C62.35C23  C62.40C23  C12.30D12  C62.43C23  C62.53C23  C62.58C23  C62.65C23  C62.70C23  C12⋊Dic6  Dic18⋊S3  C12.85S32  C335(C2×Q8)  C336(C2×Q8)  C329(S3×Q8)  C3⋊S34Dic6

Matrix representation of Dic3.D6 in GL4(𝔽5) generated by

0004
0020
0210
1001
,
0030
0004
3000
0100
,
0003
0340
0400
2002
,
0040
2002
1000
0210
G:=sub<GL(4,GF(5))| [0,0,0,1,0,0,2,0,0,2,1,0,4,0,0,1],[0,0,3,0,0,0,0,1,3,0,0,0,0,4,0,0],[0,0,0,2,0,3,4,0,0,4,0,0,3,0,0,2],[0,2,1,0,0,0,0,2,4,0,0,1,0,2,0,0] >;

Dic3.D6 in GAP, Magma, Sage, TeX

{\rm Dic}_3.D_6
% in TeX

G:=Group("Dic3.D6");
// GroupNames label

G:=SmallGroup(144,140);
// by ID

G=gap.SmallGroup(144,140);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,55,218,116,50,490,3461]);
// Polycyclic

G:=Group<a,b,c,d|a^6=1,b^2=c^6=d^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^5>;
// generators/relations

Export

Character table of Dic3.D6 in TeX

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