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G = C2≀C4⋊C2order 128 = 27

6th semidirect product of C2≀C4 and C2 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C2≀C46C2, C22≀C23C4, C23.5(C2×D4), C42⋊C27C4, (C2×D4).127D4, (C22×D4)⋊10C4, C24.13(C2×C4), (C2×Q8).115D4, (C22×C4).92D4, C4.16(C23⋊C4), (C2×D4).16C23, C23⋊C4.9C22, C22≀C2.2C22, C23.55(C22×C4), C23.20(C22⋊C4), C22.29C24.6C2, C4.D4.10C22, C23.C2313C2, M4(2).8C2215C2, (C2×C4).5(C2×D4), C22⋊C4.2(C2×C4), C2.34(C2×C23⋊C4), (C2×D4).125(C2×C4), (C22×C4).29(C2×C4), (C2×C4○D4).72C22, C22.58(C2×C22⋊C4), (C2×C4).144(C22⋊C4), SmallGroup(128,854)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C2≀C4⋊C2
C1C2C22C23C2×D4C2×C4○D4C22.29C24 — C2≀C4⋊C2
C1C2C22C23 — C2≀C4⋊C2
C1C2C2×C4C2×C4○D4 — C2≀C4⋊C2
C1C2C22C2×D4 — C2≀C4⋊C2

Generators and relations for C2≀C4⋊C2
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, eae-1=abcd, faf=acd, bc=cb, fbf=bd=db, ebe-1=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=cde >

Subgroups: 380 in 133 conjugacy classes, 42 normal (26 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×7], C22, C22 [×16], C8 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×11], Q8, C23 [×3], C23 [×6], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C2×C8, M4(2) [×3], C22×C4 [×3], C2×D4 [×3], C2×D4 [×8], C2×Q8, C4○D4 [×2], C24 [×2], C23⋊C4 [×2], C23⋊C4, C4.D4 [×2], C4.10D4, C42⋊C2, C42⋊C2, C22≀C2 [×4], C4⋊D4 [×2], C4.4D4, C41D4, C2×M4(2), C22×D4, C2×C4○D4, C2≀C4 [×4], C23.C23, M4(2).8C22, C22.29C24, C2≀C4⋊C2
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C23⋊C4 [×2], C2×C22⋊C4, C2×C23⋊C4, C2≀C4⋊C2

Character table of C2≀C4⋊C2

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11244488224448888888888
ρ111111111111111111111111    trivial
ρ2111111-1-1111111-1111-1-1-1-1-1    linear of order 2
ρ31111-11-11-1-11-1-1-1-11-111-1-111    linear of order 2
ρ41111-111-1-1-11-1-1-111-11-111-1-1    linear of order 2
ρ51111-111-1-1-11-1-111-11-1-1-1-111    linear of order 2
ρ61111-11-11-1-11-1-11-1-11-1111-1-1    linear of order 2
ρ7111111-1-111111-1-1-1-1-1-11111    linear of order 2
ρ81111111111111-11-1-1-11-1-1-1-1    linear of order 2
ρ91111-1-1-1-111-1-11-i1ii-i1-iii-i    linear of order 4
ρ101111-1-11111-1-11-i-1ii-i-1i-i-ii    linear of order 4
ρ1111111-11-1-1-1-11-1i-1i-i-i1i-ii-i    linear of order 4
ρ1211111-1-11-1-1-11-1i1i-i-i-1-ii-ii    linear of order 4
ρ1311111-1-11-1-1-11-1-i1-iii-1i-ii-i    linear of order 4
ρ1411111-11-1-1-1-11-1-i-1-iii1-ii-ii    linear of order 4
ρ151111-1-11111-1-11i-1-i-ii-1-iii-i    linear of order 4
ρ161111-1-1-1-111-1-11i1-i-ii1i-i-ii    linear of order 4
ρ17222-2-220022-22-20000000000    orthogonal lifted from D4
ρ18222-22200-2-2-2-220000000000    orthogonal lifted from D4
ρ19222-2-2-200-2-22220000000000    orthogonal lifted from D4
ρ20222-22-200222-2-20000000000    orthogonal lifted from D4
ρ2144-400000-440000000000000    orthogonal lifted from C23⋊C4
ρ2244-4000004-40000000000000    orthogonal lifted from C23⋊C4
ρ238-8000000000000000000000    orthogonal faithful

Permutation representations of C2≀C4⋊C2
On 16 points - transitive group 16T213
Generators in S16
(1 7)(2 14)(3 5)(8 10)(9 13)(11 15)
(1 7)(2 10)(3 11)(4 6)(5 15)(8 14)(9 13)(12 16)
(2 14)(4 16)(6 12)(8 10)
(1 13)(2 14)(3 15)(4 16)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(2 14)(3 15)(6 12)(7 9)

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

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

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

G:=TransitiveGroup(16,213);

On 16 points - transitive group 16T281
Generators in S16
(1 13)(3 9)(4 8)(6 15)(7 11)(10 12)
(1 13)(2 14)(3 11)(4 12)(5 16)(6 15)(7 9)(8 10)
(1 6)(3 7)(9 11)(13 15)
(1 6)(2 5)(3 7)(4 8)(9 11)(10 12)(13 15)(14 16)
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)
(1 11)(2 12)(3 15)(4 16)(5 10)(6 9)(7 13)(8 14)

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

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

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

G:=TransitiveGroup(16,281);

On 16 points - transitive group 16T315
Generators in S16
(4 14)(5 11)(6 12)(7 9)
(1 15)(4 14)(5 11)(8 10)
(2 16)(4 14)(6 12)(8 10)
(1 15)(2 16)(3 13)(4 14)(5 11)(6 12)(7 9)(8 10)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 7)(2 10)(3 11)(4 6)(5 13)(8 16)(9 15)(12 14)

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

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

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

G:=TransitiveGroup(16,315);

Matrix representation of C2≀C4⋊C2 in GL8(ℤ)

00010000
00100000
01000000
10000000
00000100
00001000
00000001
00000010
,
00-100000
000-10000
-10000000
0-1000000
00000010
00000001
00001000
00000100
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
-10000000
0-1000000
00-100000
000-10000
0000-1000
00000-100
000000-10
0000000-1
,
0000000-1
00000010
00000-100
00001000
00100000
000-10000
-10000000
01000000
,
10000000
0-1000000
00-100000
00010000
0000-1000
00000100
00000010
0000000-1

G:=sub<GL(8,Integers())| [0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1] >;

C2≀C4⋊C2 in GAP, Magma, Sage, TeX

C_2\wr C_4\rtimes C_2
% in TeX

G:=Group("C2wrC4:C2");
// GroupNames label

G:=SmallGroup(128,854);
// by ID

G=gap.SmallGroup(128,854);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,723,352,1123,851,375,4037]);
// Polycyclic

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

Export

Character table of C2≀C4⋊C2 in TeX

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