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G = D4×C32⋊C4order 288 = 25·32

Direct product of D4 and C32⋊C4

direct product, metabelian, soluble, monomial

Aliases: D4×C32⋊C4, C62⋊(C2×C4), C327D4⋊C4, C12⋊S33C4, C3212(C4×D4), (D4×C32)⋊3C4, C62⋊C46C2, (C3×C12)⋊(C2×C4), C41(C2×C32⋊C4), (C4×C32⋊C4)⋊5C2, (D4×C3⋊S3).4C2, C4⋊(C32⋊C4)⋊5C2, C3⋊Dic34(C2×C4), C3⋊S3.11(C2×D4), C221(C2×C32⋊C4), (C22×C32⋊C4)⋊3C2, C2.9(C22×C32⋊C4), C3⋊S3.11(C4○D4), (C2×C3⋊S3).37C23, (C4×C3⋊S3).38C22, (C3×C6).31(C22×C4), (C2×C32⋊C4).24C22, (C22×C3⋊S3).56C22, (C2×C3⋊S3)⋊6(C2×C4), SmallGroup(288,936)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D4×C32⋊C4
C1C32C3⋊S3C2×C3⋊S3C2×C32⋊C4C22×C32⋊C4 — D4×C32⋊C4
C32C3×C6 — D4×C32⋊C4
C1C2D4

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 12A2B2C2D2E2F2G3A3B4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E6F12A12B
 size 112299181844299991818181818181844888888
ρ1111111111111111111111111111111    trivial
ρ211-1-111-1-1111-1-1-1-1-111111-111-1-1-1-111    linear of order 2
ρ3111-1111-111-1-1-1-1-11-111-1-1111-111-1-1-1    linear of order 2
ρ411-1111-1111-11111-1-111-1-1-1111-1-11-1-1    linear of order 2
ρ511111111111-1-1-1-1-11-1-1-1-1-111111111    linear of order 2
ρ611-1-111-1-1111111111-1-1-1-1111-1-1-1-111    linear of order 2
ρ7111-1111-111-11111-1-1-1-111-111-111-1-1-1    linear of order 2
ρ811-1111-1111-1-1-1-1-11-1-1-1111111-1-11-1-1    linear of order 2
ρ911-1-1-1-111111-iii-i-i-1-ii-iii11-1-1-1-111    linear of order 4
ρ1011-1-1-1-111111i-i-iii-1i-ii-i-i11-1-1-1-111    linear of order 4
ρ1111-11-1-11-111-1-iii-ii1i-i-ii-i111-1-11-1-1    linear of order 4
ρ1211-11-1-11-111-1i-i-ii-i1-iii-ii111-1-11-1-1    linear of order 4
ρ131111-1-1-1-1111-iii-i-i-1i-ii-ii11111111    linear of order 4
ρ141111-1-1-1-1111i-i-iii-1-ii-ii-i11111111    linear of order 4
ρ15111-1-1-1-1111-1-iii-ii1-iii-i-i11-111-1-1-1    linear of order 4
ρ16111-1-1-1-1111-1i-i-ii-i1i-i-iii11-111-1-1-1    linear of order 4
ρ172-2002-200220-22-220000000-2-2000000    orthogonal lifted from D4
ρ182-2002-2002202-22-20000000-2-2000000    orthogonal lifted from D4
ρ192-200-2200220-2i-2i2i2i0000000-2-2000000    complex lifted from C4○D4
ρ202-200-22002202i2i-2i-2i0000000-2-2000000    complex lifted from C4○D4
ρ21444-400001-2-400000000000-212-21-12-1    orthogonal lifted from C2×C32⋊C4
ρ2244-4400001-2-400000000000-21-22-112-1    orthogonal lifted from C2×C32⋊C4
ρ23444400001-2400000000000-21-2-211-21    orthogonal lifted from C32⋊C4
ρ2444-4-400001-2400000000000-2122-1-1-21    orthogonal lifted from C2×C32⋊C4
ρ25444-40000-21-4000000000001-2-11-22-12    orthogonal lifted from C2×C32⋊C4
ρ2644-440000-21-4000000000001-21-12-2-12    orthogonal lifted from C2×C32⋊C4
ρ2744440000-214000000000001-211-2-21-2    orthogonal lifted from C32⋊C4
ρ2844-4-40000-214000000000001-2-1-1221-2    orthogonal lifted from C2×C32⋊C4
ρ298-8000000-42000000000000-24000000    orthogonal faithful
ρ308-80000002-40000000000004-2000000    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)

-120000
-110000
00-1000
000-100
0000-10
00000-1
,
100000
1-10000
001000
000100
000010
000001
,
100000
010000
001000
000100
000001
0000-1-1
,
100000
010000
00-1-100
001000
000001
0000-1-1
,
100000
010000
000010
000001
001000
00-1-100

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

Export

Character table of D4×C32⋊C4 in TeX

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