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G = C422D4order 128 = 27

2nd semidirect product of C42 and D4 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C422D4, C24.82D4, (C2×C8)⋊2D4, (C2×D4)⋊9D4, (C2×Q8)⋊10D4, C4.9C424C2, C4.65C22≀C2, C4.57(C41D4), D8⋊C222C2, C24.4C41C2, (C22×C4).140D4, C23.139(C2×D4), C4.141(C4⋊D4), C22.19C241C2, C22.39C22≀C2, C42⋊C2218C2, C22.24(C4⋊D4), C2.15(C232D4), (C23×C4).270C22, (C22×C4).714C23, C42⋊C2.52C22, (C2×M4(2)).19C22, (C2×C4).253(C2×D4), (C2×C4).339(C4○D4), (C2×C4○D4).51C22, SmallGroup(128,742)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C422D4
C1C2C22C23C22×C4C23×C4C22.19C24 — C422D4
C1C2C22×C4 — C422D4
C1C4C22×C4 — C422D4
C1C2C2C22×C4 — C422D4

Generators and relations for C422D4
 G = < a,b,c,d | a4=b4=c4=d2=1, dad=ab=ba, cac-1=a-1b, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 448 in 197 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×7], C4 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×17], C8 [×3], C2×C4 [×2], C2×C4 [×4], C2×C4 [×17], D4 [×16], Q8 [×4], C23, C23 [×7], C42 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×2], C2×C8, M4(2) [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×8], C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×8], C24, C22⋊C8 [×2], C4≀C2 [×4], C42⋊C2 [×2], C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C2×M4(2) [×2], C4○D8 [×2], C8⋊C22 [×2], C8.C22 [×2], C23×C4, C2×C4○D4 [×2], C4.9C42, C24.4C4, C42⋊C22 [×2], C22.19C24 [×2], D8⋊C22, C422D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×12], C23, C2×D4 [×6], C4○D4, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C232D4, C422D4

Character table of C422D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11222448811222448888888888
ρ111111111111111111111111111    trivial
ρ21111111-1-11111111-1-1-1-1-1-11111    linear of order 2
ρ311111-1-11111111-1-1-1-11-1-1111-1-1    linear of order 2
ρ411111-1-1-1-111111-1-111-111-111-1-1    linear of order 2
ρ511111-1-11-111111-1-1-111-11-1-1-111    linear of order 2
ρ611111-1-1-1111111-1-11-1-11-11-1-111    linear of order 2
ρ711111111-111111111-111-1-1-1-1-1-1    linear of order 2
ρ81111111-111111111-11-1-111-1-1-1-1    linear of order 2
ρ922-22-2000022-2-2200000000-2200    orthogonal lifted from D4
ρ1022-2-22000-2-2-2-222000000020000    orthogonal lifted from D4
ρ11222-2-20020-2-22-220000-20000000    orthogonal lifted from D4
ρ12222-2-200-20-2-22-22000020000000    orthogonal lifted from D4
ρ1322-22-2000022-2-22000000002-200    orthogonal lifted from D4
ρ1422222-2-200-2-2-2-2-2220000000000    orthogonal lifted from D4
ρ1522-2-220000222-2-2000200-200000    orthogonal lifted from D4
ρ16222-2-2000022-22-200-2002000000    orthogonal lifted from D4
ρ17222-2-2000022-22-200200-2000000    orthogonal lifted from D4
ρ18222222200-2-2-2-2-2-2-20000000000    orthogonal lifted from D4
ρ1922-2-220000222-2-2000-200200000    orthogonal lifted from D4
ρ2022-2-220002-2-2-2220000000-20000    orthogonal lifted from D4
ρ2122-22-20000-2-222-200000000002i-2i    complex lifted from C4○D4
ρ2222-22-20000-2-222-20000000000-2i2i    complex lifted from C4○D4
ρ234-40002-2004i-4i000-2i2i0000000000    complex faithful
ρ244-4000-2200-4i4i000-2i2i0000000000    complex faithful
ρ254-4000-22004i-4i0002i-2i0000000000    complex faithful
ρ264-40002-200-4i4i0002i-2i0000000000    complex faithful

Permutation representations of C422D4
On 16 points - transitive group 16T382
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 3 4 2)(5 7 6 8)(9 10 11 12)(13 16 15 14)
(1 14 7 12)(2 13 5 9)(3 15 6 11)(4 16 8 10)
(1 16)(2 15)(3 13)(4 14)(5 11)(6 9)(7 10)(8 12)

G:=sub<Sym(16)| (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,4,2)(5,7,6,8)(9,10,11,12)(13,16,15,14), (1,14,7,12)(2,13,5,9)(3,15,6,11)(4,16,8,10), (1,16)(2,15)(3,13)(4,14)(5,11)(6,9)(7,10)(8,12)>;

G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,4,2)(5,7,6,8)(9,10,11,12)(13,16,15,14), (1,14,7,12)(2,13,5,9)(3,15,6,11)(4,16,8,10), (1,16)(2,15)(3,13)(4,14)(5,11)(6,9)(7,10)(8,12) );

G=PermutationGroup([(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,4,2),(5,7,6,8),(9,10,11,12),(13,16,15,14)], [(1,14,7,12),(2,13,5,9),(3,15,6,11),(4,16,8,10)], [(1,16),(2,15),(3,13),(4,14),(5,11),(6,9),(7,10),(8,12)])

G:=TransitiveGroup(16,382);

On 16 points - transitive group 16T406
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 6 7 3)(2 5 8 4)(9 15 11 13)(10 16 12 14)
(1 10)(2 13 8 15)(3 16)(4 9 5 11)(6 14)(7 12)
(1 10)(2 13)(3 16)(4 9)(5 11)(6 14)(7 12)(8 15)

G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,6,7,3)(2,5,8,4)(9,15,11,13)(10,16,12,14), (1,10)(2,13,8,15)(3,16)(4,9,5,11)(6,14)(7,12), (1,10)(2,13)(3,16)(4,9)(5,11)(6,14)(7,12)(8,15)>;

G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,6,7,3)(2,5,8,4)(9,15,11,13)(10,16,12,14), (1,10)(2,13,8,15)(3,16)(4,9,5,11)(6,14)(7,12), (1,10)(2,13)(3,16)(4,9)(5,11)(6,14)(7,12)(8,15) );

G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,6,7,3),(2,5,8,4),(9,15,11,13),(10,16,12,14)], [(1,10),(2,13,8,15),(3,16),(4,9,5,11),(6,14),(7,12)], [(1,10),(2,13),(3,16),(4,9),(5,11),(6,14),(7,12),(8,15)])

G:=TransitiveGroup(16,406);

Matrix representation of C422D4 in GL4(𝔽5) generated by

2000
0400
0010
0003
,
3000
0200
0020
0003
,
0010
1000
0001
0100
,
0010
0001
1000
0100
G:=sub<GL(4,GF(5))| [2,0,0,0,0,4,0,0,0,0,1,0,0,0,0,3],[3,0,0,0,0,2,0,0,0,0,2,0,0,0,0,3],[0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;

C422D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2D_4
% in TeX

G:=Group("C4^2:2D4");
// GroupNames label

G:=SmallGroup(128,742);
// by ID

G=gap.SmallGroup(128,742);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,2019,1018,248,1411,4037]);
// Polycyclic

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

Export

Character table of C422D4 in TeX

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