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## G = D4×C3⋊S3order 144 = 24·32

### Direct product of D4 and C3⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D4×C3⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — D4×C3⋊S3
 Lower central C32 — C3×C6 — D4×C3⋊S3
 Upper central C1 — C2 — D4

Generators and relations for D4×C3⋊S3
G = < a,b,c,d,e | a4=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >

Subgroups: 610 in 162 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2 [×6], C3 [×4], C4, C4, C22 [×2], C22 [×7], S3 [×16], C6 [×4], C6 [×8], C2×C4, D4, D4 [×3], C23 [×2], C32, Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×8], C2×D4, C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×4], D12 [×4], C3⋊D4 [×8], C3×D4 [×4], C22×S3 [×8], C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62 [×2], S3×D4 [×4], C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C22×C3⋊S3 [×2], D4×C3⋊S3
Quotients: C1, C2 [×7], C22 [×7], S3 [×4], D4 [×2], C23, D6 [×12], C2×D4, C3⋊S3, C22×S3 [×4], C2×C3⋊S3 [×3], S3×D4 [×4], C22×C3⋊S3, D4×C3⋊S3

Character table of D4×C3⋊S3

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 12A 12B 12C 12D size 1 1 2 2 9 9 18 18 2 2 2 2 2 18 2 2 2 2 4 4 4 4 4 4 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 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 -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 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 -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 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 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 -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 -1 -1 -1 -1 -1 -1 linear of order 2 ρ9 2 2 2 -2 0 0 0 0 -1 -1 2 -1 -2 0 -1 -1 2 -1 1 -1 -1 -1 2 1 1 -2 1 -2 1 1 orthogonal lifted from D6 ρ10 2 2 -2 -2 0 0 0 0 -1 -1 -1 2 2 0 2 -1 -1 -1 1 1 -2 1 1 -2 1 1 -1 -1 -1 2 orthogonal lifted from D6 ρ11 2 2 2 2 0 0 0 0 -1 -1 -1 2 2 0 2 -1 -1 -1 -1 -1 2 -1 -1 2 -1 -1 -1 -1 -1 2 orthogonal lifted from S3 ρ12 2 2 -2 2 0 0 0 0 -1 -1 -1 2 -2 0 2 -1 -1 -1 -1 1 -2 1 1 2 -1 -1 1 1 1 -2 orthogonal lifted from D6 ρ13 2 2 2 -2 0 0 0 0 -1 -1 -1 2 -2 0 2 -1 -1 -1 1 -1 2 -1 -1 -2 1 1 1 1 1 -2 orthogonal lifted from D6 ρ14 2 2 2 2 0 0 0 0 -1 -1 2 -1 2 0 -1 -1 2 -1 -1 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 orthogonal lifted from S3 ρ15 2 2 2 2 0 0 0 0 2 -1 -1 -1 2 0 -1 -1 -1 2 2 2 -1 -1 -1 -1 -1 -1 -1 -1 2 -1 orthogonal lifted from S3 ρ16 2 2 -2 -2 0 0 0 0 2 -1 -1 -1 2 0 -1 -1 -1 2 -2 -2 1 1 1 1 1 1 -1 -1 2 -1 orthogonal lifted from D6 ρ17 2 -2 0 0 -2 2 0 0 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 0 0 0 0 2 -1 -1 -1 -2 0 -1 -1 -1 2 -2 2 -1 -1 -1 1 1 1 1 1 -2 1 orthogonal lifted from D6 ρ19 2 2 -2 2 0 0 0 0 2 -1 -1 -1 -2 0 -1 -1 -1 2 2 -2 1 1 1 -1 -1 -1 1 1 -2 1 orthogonal lifted from D6 ρ20 2 2 -2 2 0 0 0 0 -1 -1 2 -1 -2 0 -1 -1 2 -1 -1 1 1 1 -2 -1 -1 2 1 -2 1 1 orthogonal lifted from D6 ρ21 2 2 -2 -2 0 0 0 0 -1 -1 2 -1 2 0 -1 -1 2 -1 1 1 1 1 -2 1 1 -2 -1 2 -1 -1 orthogonal lifted from D6 ρ22 2 -2 0 0 2 -2 0 0 2 2 2 2 0 0 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 2 2 0 0 0 0 -1 2 -1 -1 2 0 -1 2 -1 -1 -1 -1 -1 2 -1 -1 2 -1 2 -1 -1 -1 orthogonal lifted from S3 ρ24 2 2 -2 -2 0 0 0 0 -1 2 -1 -1 2 0 -1 2 -1 -1 1 1 1 -2 1 1 -2 1 2 -1 -1 -1 orthogonal lifted from D6 ρ25 2 2 2 -2 0 0 0 0 -1 2 -1 -1 -2 0 -1 2 -1 -1 1 -1 -1 2 -1 1 -2 1 -2 1 1 1 orthogonal lifted from D6 ρ26 2 2 -2 2 0 0 0 0 -1 2 -1 -1 -2 0 -1 2 -1 -1 -1 1 1 -2 1 -1 2 -1 -2 1 1 1 orthogonal lifted from D6 ρ27 4 -4 0 0 0 0 0 0 -2 4 -2 -2 0 0 2 -4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ28 4 -4 0 0 0 0 0 0 4 -2 -2 -2 0 0 2 2 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ29 4 -4 0 0 0 0 0 0 -2 -2 4 -2 0 0 2 2 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ30 4 -4 0 0 0 0 0 0 -2 -2 -2 4 0 0 -4 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4

Smallest permutation representation of D4×C3⋊S3
On 36 points
Generators in S36
(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)
(1 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 22)(23 24)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(1 7 29)(2 8 30)(3 5 31)(4 6 32)(9 16 27)(10 13 28)(11 14 25)(12 15 26)(17 23 36)(18 24 33)(19 21 34)(20 22 35)
(1 24 16)(2 21 13)(3 22 14)(4 23 15)(5 35 25)(6 36 26)(7 33 27)(8 34 28)(9 29 18)(10 30 19)(11 31 20)(12 32 17)
(1 3)(2 4)(5 29)(6 30)(7 31)(8 32)(9 35)(10 36)(11 33)(12 34)(13 23)(14 24)(15 21)(16 22)(17 28)(18 25)(19 26)(20 27)

G:=sub<Sym(36)| (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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,35,25)(6,36,26)(7,33,27)(8,34,28)(9,29,18)(10,30,19)(11,31,20)(12,32,17), (1,3)(2,4)(5,29)(6,30)(7,31)(8,32)(9,35)(10,36)(11,33)(12,34)(13,23)(14,24)(15,21)(16,22)(17,28)(18,25)(19,26)(20,27)>;

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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,7,29)(2,8,30)(3,5,31)(4,6,32)(9,16,27)(10,13,28)(11,14,25)(12,15,26)(17,23,36)(18,24,33)(19,21,34)(20,22,35), (1,24,16)(2,21,13)(3,22,14)(4,23,15)(5,35,25)(6,36,26)(7,33,27)(8,34,28)(9,29,18)(10,30,19)(11,31,20)(12,32,17), (1,3)(2,4)(5,29)(6,30)(7,31)(8,32)(9,35)(10,36)(11,33)(12,34)(13,23)(14,24)(15,21)(16,22)(17,28)(18,25)(19,26)(20,27) );

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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,22),(23,24),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)], [(1,7,29),(2,8,30),(3,5,31),(4,6,32),(9,16,27),(10,13,28),(11,14,25),(12,15,26),(17,23,36),(18,24,33),(19,21,34),(20,22,35)], [(1,24,16),(2,21,13),(3,22,14),(4,23,15),(5,35,25),(6,36,26),(7,33,27),(8,34,28),(9,29,18),(10,30,19),(11,31,20),(12,32,17)], [(1,3),(2,4),(5,29),(6,30),(7,31),(8,32),(9,35),(10,36),(11,33),(12,34),(13,23),(14,24),(15,21),(16,22),(17,28),(18,25),(19,26),(20,27)])

Matrix representation of D4×C3⋊S3 in GL6(ℤ)

 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -2 0 0 0 0 1 1
,
 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 -2 0 0 0 0 0 1
,
 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 -1 1 0 0 0 0 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-2,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

D4×C3⋊S3 in GAP, Magma, Sage, TeX

D_4\times C_3\rtimes S_3
% in TeX

G:=Group("D4xC3:S3");
// GroupNames label

G:=SmallGroup(144,172);
// by ID

G=gap.SmallGroup(144,172);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,964,3461]);
// Polycyclic

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

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