p-group, metabelian, nilpotent (class 4), monomial
Aliases: C2≀C4⋊6C2, C22≀C2⋊3C4, C23.5(C2×D4), C42⋊C2⋊7C4, (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.C23⋊13C2, M4(2).8C22⋊15C2, (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
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, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C23⋊C4, C23⋊C4, C4.D4, C4.10D4, C42⋊C2, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C2×M4(2), C22×D4, C2×C4○D4, C2≀C4, C23.C23, M4(2).8C22, C22.29C24, C2≀C4⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C23⋊C4, C2×C22⋊C4, C2×C23⋊C4, C2≀C4⋊C2
Character table of C2≀C4⋊C2
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 8A | 8B | 8C | 8D | |
size | 1 | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | 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 | linear of order 2 |
ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | 1 | i | i | -i | 1 | -i | i | i | -i | linear of order 4 |
ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | -1 | i | i | -i | -1 | i | -i | -i | i | linear of order 4 |
ρ11 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | i | -1 | i | -i | -i | 1 | i | -i | i | -i | linear of order 4 |
ρ12 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | i | 1 | i | -i | -i | -1 | -i | i | -i | i | linear of order 4 |
ρ13 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -i | 1 | -i | i | i | -1 | i | -i | i | -i | linear of order 4 |
ρ14 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -i | -1 | -i | i | i | 1 | -i | i | -i | i | linear of order 4 |
ρ15 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | i | -1 | -i | -i | i | -1 | -i | i | i | -i | linear of order 4 |
ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | i | 1 | -i | -i | i | 1 | i | -i | -i | i | linear of order 4 |
ρ17 | 2 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ18 | 2 | 2 | 2 | -2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ19 | 2 | 2 | 2 | -2 | -2 | -2 | 0 | 0 | -2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ20 | 2 | 2 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
ρ21 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ22 | 4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23⋊C4 |
ρ23 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 7)(2 16)(3 5)(8 10)(9 15)(11 13)
(1 7)(2 10)(3 11)(4 6)(5 13)(8 16)(9 15)(12 14)
(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)
(2 16)(3 13)(6 12)(7 9)
G:=sub<Sym(16)| (1,7)(2,16)(3,5)(8,10)(9,15)(11,13), (1,7)(2,10)(3,11)(4,6)(5,13)(8,16)(9,15)(12,14), (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), (2,16)(3,13)(6,12)(7,9)>;
G:=Group( (1,7)(2,16)(3,5)(8,10)(9,15)(11,13), (1,7)(2,10)(3,11)(4,6)(5,13)(8,16)(9,15)(12,14), (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), (2,16)(3,13)(6,12)(7,9) );
G=PermutationGroup([[(1,7),(2,16),(3,5),(8,10),(9,15),(11,13)], [(1,7),(2,10),(3,11),(4,6),(5,13),(8,16),(9,15),(12,14)], [(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)], [(2,16),(3,13),(6,12),(7,9)]])
G:=TransitiveGroup(16,213);
(1 13)(3 8)(4 9)(6 15)(7 11)(10 12)
(1 13)(2 14)(3 12)(4 11)(5 16)(6 15)(7 9)(8 10)
(1 6)(4 7)(9 11)(13 15)
(1 6)(2 5)(3 8)(4 7)(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 16)(4 15)(5 10)(6 9)(7 13)(8 14)
G:=sub<Sym(16)| (1,13)(3,8)(4,9)(6,15)(7,11)(10,12), (1,13)(2,14)(3,12)(4,11)(5,16)(6,15)(7,9)(8,10), (1,6)(4,7)(9,11)(13,15), (1,6)(2,5)(3,8)(4,7)(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)(3,16)(4,15)(5,10)(6,9)(7,13)(8,14)>;
G:=Group( (1,13)(3,8)(4,9)(6,15)(7,11)(10,12), (1,13)(2,14)(3,12)(4,11)(5,16)(6,15)(7,9)(8,10), (1,6)(4,7)(9,11)(13,15), (1,6)(2,5)(3,8)(4,7)(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)(3,16)(4,15)(5,10)(6,9)(7,13)(8,14) );
G=PermutationGroup([[(1,13),(3,8),(4,9),(6,15),(7,11),(10,12)], [(1,13),(2,14),(3,12),(4,11),(5,16),(6,15),(7,9),(8,10)], [(1,6),(4,7),(9,11),(13,15)], [(1,6),(2,5),(3,8),(4,7),(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),(3,16),(4,15),(5,10),(6,9),(7,13),(8,14)]])
G:=TransitiveGroup(16,281);
(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)
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 | 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 | 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 |
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 |
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
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