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## G = D4×C32⋊C4order 288 = 25·32

### Direct product of D4 and C32⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — D4×C32⋊C4
 Chief series C1 — C32 — C3⋊S3 — C2×C3⋊S3 — C2×C32⋊C4 — C22×C32⋊C4 — D4×C32⋊C4
 Lower central C32 — C3×C6 — D4×C32⋊C4
 Upper central C1 — C2 — D4

Generators and relations for D4×C32⋊C4
G = < a,b,c,d,e | a4=b2=c3=d3=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ede-1=cd=dc, ece-1=c-1d >

Subgroups: 880 in 148 conjugacy classes, 38 normal (20 characteristic)
C1, C2, C2 [×6], C3 [×2], C4, C4 [×6], C22 [×2], C22 [×7], S3 [×8], C6 [×6], C2×C4 [×9], D4, D4 [×3], C23 [×2], C32, Dic3 [×2], C12 [×2], D6 [×14], C2×C6 [×4], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3⋊S3 [×2], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×4], C3×D4 [×2], C22×S3 [×4], C4×D4, C3⋊Dic3, C3×C12, C32⋊C4 [×2], C32⋊C4 [×3], C2×C3⋊S3, C2×C3⋊S3 [×2], C2×C3⋊S3 [×4], C62 [×2], S3×D4 [×2], C4×C3⋊S3, C12⋊S3, C327D4 [×2], D4×C32, C2×C32⋊C4 [×2], C2×C32⋊C4 [×2], C2×C32⋊C4 [×4], C22×C3⋊S3 [×2], C4×C32⋊C4, C4⋊(C32⋊C4), C62⋊C4 [×2], D4×C3⋊S3, C22×C32⋊C4 [×2], D4×C32⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, C22×C4, C2×D4, C4○D4, C4×D4, C32⋊C4, C2×C32⋊C4 [×3], C22×C32⋊C4, D4×C32⋊C4

Character table of D4×C32⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 6F 12A 12B size 1 1 2 2 9 9 18 18 4 4 2 9 9 9 9 18 18 18 18 18 18 18 4 4 8 8 8 8 8 8 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 1 1 -1 -1 -1 -1 1 1 1 1 1 -i i i -i -i -1 -i i -i i i 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ10 1 1 -1 -1 -1 -1 1 1 1 1 1 i -i -i i i -1 i -i i -i -i 1 1 -1 -1 -1 -1 1 1 linear of order 4 ρ11 1 1 -1 1 -1 -1 1 -1 1 1 -1 -i i i -i i 1 i -i -i i -i 1 1 1 -1 -1 1 -1 -1 linear of order 4 ρ12 1 1 -1 1 -1 -1 1 -1 1 1 -1 i -i -i i -i 1 -i i i -i i 1 1 1 -1 -1 1 -1 -1 linear of order 4 ρ13 1 1 1 1 -1 -1 -1 -1 1 1 1 -i i i -i -i -1 i -i i -i i 1 1 1 1 1 1 1 1 linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 1 1 1 i -i -i i i -1 -i i -i i -i 1 1 1 1 1 1 1 1 linear of order 4 ρ15 1 1 1 -1 -1 -1 -1 1 1 1 -1 -i i i -i i 1 -i i i -i -i 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ16 1 1 1 -1 -1 -1 -1 1 1 1 -1 i -i -i i -i 1 i -i -i i i 1 1 -1 1 1 -1 -1 -1 linear of order 4 ρ17 2 -2 0 0 2 -2 0 0 2 2 0 -2 2 -2 2 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 0 0 2 -2 0 0 2 2 0 2 -2 2 -2 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 0 0 -2 2 0 0 2 2 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 0 0 -2 2 0 0 2 2 0 2i 2i -2i -2i 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 4 4 4 -4 0 0 0 0 1 -2 -4 0 0 0 0 0 0 0 0 0 0 0 -2 1 2 -2 1 -1 2 -1 orthogonal lifted from C2×C32⋊C4 ρ22 4 4 -4 4 0 0 0 0 1 -2 -4 0 0 0 0 0 0 0 0 0 0 0 -2 1 -2 2 -1 1 2 -1 orthogonal lifted from C2×C32⋊C4 ρ23 4 4 4 4 0 0 0 0 1 -2 4 0 0 0 0 0 0 0 0 0 0 0 -2 1 -2 -2 1 1 -2 1 orthogonal lifted from C32⋊C4 ρ24 4 4 -4 -4 0 0 0 0 1 -2 4 0 0 0 0 0 0 0 0 0 0 0 -2 1 2 2 -1 -1 -2 1 orthogonal lifted from C2×C32⋊C4 ρ25 4 4 4 -4 0 0 0 0 -2 1 -4 0 0 0 0 0 0 0 0 0 0 0 1 -2 -1 1 -2 2 -1 2 orthogonal lifted from C2×C32⋊C4 ρ26 4 4 -4 4 0 0 0 0 -2 1 -4 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 -1 2 -2 -1 2 orthogonal lifted from C2×C32⋊C4 ρ27 4 4 4 4 0 0 0 0 -2 1 4 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 1 -2 -2 1 -2 orthogonal lifted from C32⋊C4 ρ28 4 4 -4 -4 0 0 0 0 -2 1 4 0 0 0 0 0 0 0 0 0 0 0 1 -2 -1 -1 2 2 1 -2 orthogonal lifted from C2×C32⋊C4 ρ29 8 -8 0 0 0 0 0 0 -4 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 4 0 0 0 0 0 0 orthogonal faithful ρ30 8 -8 0 0 0 0 0 0 2 -4 0 0 0 0 0 0 0 0 0 0 0 0 4 -2 0 0 0 0 0 0 orthogonal faithful

Permutation representations of D4×C32⋊C4
On 24 points - transitive group 24T618
Generators in S24
(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 3)(5 6)(7 8)(9 12)(10 11)(13 14)(15 16)(17 20)(18 19)(21 22)(23 24)
(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(1 24)(2 21)(3 22)(4 23)(5 15 10 20)(6 16 11 17)(7 13 12 18)(8 14 9 19)

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

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

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

G:=TransitiveGroup(24,618);

On 24 points - transitive group 24T619
Generators in S24
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(2 4)(5 7)(10 12)(13 15)(18 20)(21 23)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 23 10)(6 24 11)(7 21 12)(8 22 9)
(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 22 3 24)(2 23 4 21)(5 15 12 18)(6 16 9 19)(7 13 10 20)(8 14 11 17)

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

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

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

G:=TransitiveGroup(24,619);

Matrix representation of D4×C32⋊C4 in GL6(ℤ)

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

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

D4×C32⋊C4 in GAP, Magma, Sage, TeX

D_4\times C_3^2\rtimes C_4
% in TeX

G:=Group("D4xC3^2:C4");
// GroupNames label

G:=SmallGroup(288,936);
// by ID

G=gap.SmallGroup(288,936);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,219,9413,362,12550,1203]);
// Polycyclic

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

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