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G = C42⋊2D4  order 128 = 27

2nd semidirect product of C42 and D4 acting faithfully

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

Aliases: C42⋊2D4, C24.82D4, (C2×C8)⋊2D4, (C2×D4)⋊9D4, (C2×Q8)⋊10D4, C4.9C42⋊4C2, C4.65C22≀C2, C4.57(C4⋊1D4), D8⋊C22⋊2C2, C24.4C4⋊1C2, (C22×C4).140D4, C23.139(C2×D4), C4.141(C4⋊D4), C22.19C24⋊1C2, C22.39C22≀C2, C42⋊C22⋊18C2, C22.24(C4⋊D4), C2.15(C23⋊2D4), (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

Derived series
C1 — C22×C4 — C42⋊2D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22.19C24 — C42⋊2D4
Lower central
C1 — C2 — C22×C4 — C42⋊2D4
Upper central
C1 — C4 — C22×C4 — C42⋊2D4
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊2D4

Generators and relations for C42⋊2D4
 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, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C22⋊C8, C4≀C2, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C2×M4(2), C4○D8, C8⋊C22, C8.C22, C23×C4, C2×C4○D4, C4.9C42, C24.4C4, C42⋊C22, C22.19C24, D8⋊C22, C42⋊2D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C22≀C2, C4⋊D4, C4⋊1D4, C23⋊2D4, C42⋊2D4

Character table of C42⋊2D4

 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 C42⋊2D4
►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 8 6 7)(9 10 11 12)(13 16 15 14)

(1 14 8 11)(2 13 5 12)(3 15 6 10)(4 16 7 9)

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

(1 16)(2 12 5 10)(3 11)(4 15 7 13)(6 14)(8 9)

(1 16)(2 12)(3 11)(4 15)(5 10)(6 14)(7 13)(8 9)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,6,3)(2,7,5,4)(9,16,11,14)(10,13,12,15), (1,16)(2,12,5,10)(3,11)(4,15,7,13)(6,14)(8,9), (1,16)(2,12)(3,11)(4,15)(5,10)(6,14)(7,13)(8,9)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,8,6,3)(2,7,5,4)(9,16,11,14)(10,13,12,15), (1,16)(2,12,5,10)(3,11)(4,15,7,13)(6,14)(8,9), (1,16)(2,12)(3,11)(4,15)(5,10)(6,14)(7,13)(8,9) );
 
G=PermutationGroup([[(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,8,6,3),(2,7,5,4),(9,16,11,14),(10,13,12,15)], [(1,16),(2,12,5,10),(3,11),(4,15,7,13),(6,14),(8,9)], [(1,16),(2,12),(3,11),(4,15),(5,10),(6,14),(7,13),(8,9)]])
 
G:=TransitiveGroup(16,406);
 

Matrix representation of C42⋊2D4 ►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] >;
 

C42⋊2D4 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 C42⋊2D4 in TeX

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