Copied to
clipboard

## G = C2×C4≀C2order 64 = 26

### Direct product of C2 and C4≀C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4≀C2
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C4≀C2
 Lower central C1 — C2 — C4 — C2×C4≀C2
 Upper central C1 — C2×C4 — C22×C4 — C2×C4≀C2
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4≀C2

Generators and relations for C2×C4≀C2
G = < a,b,c,d | a2=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b-1c >

Subgroups: 137 in 85 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C42 [×2], C42, C2×C8, M4(2) [×2], M4(2), C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, 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], C4≀C2 [×2], C2×C22⋊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 4L 4M 4N 4O 4P 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 ρ1 1 1 1 1 1 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 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 -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 -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 -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 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 -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 1 1 -1 1 -1 linear of order 2 ρ9 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 i i -i i i -i -1 1 -i -i -1 1 i i -i -i linear of order 4 ρ10 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 i i -i i i -i -1 1 -i -i 1 -1 -i -i i i linear of order 4 ρ11 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -i -i i -i -i i -1 1 i i -1 1 -i -i i i linear of order 4 ρ12 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -i -i i -i -i i -1 1 i i 1 -1 i i -i -i linear of order 4 ρ13 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 i i -i -i -i -i 1 1 i i -1 -1 -i i i -i linear of order 4 ρ14 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 i i -i -i -i -i 1 1 i i 1 1 i -i -i i linear of order 4 ρ15 1 1 1 1 -1 -1 1 1 -1 -1 -1 -1 -i -i i i i i 1 1 -i -i -1 -1 i -i -i i linear of order 4 ρ16 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 -i -i i i i i 1 1 -i -i 1 1 -i i i -i linear of order 4 ρ17 2 -2 2 -2 -2 2 0 0 -2 2 2 -2 0 0 0 0 0 0 2 -2 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 0 0 0 0 0 0 -2 -2 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 0 0 0 0 0 0 2 -2 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 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 -2 2 0 0 0 0 -2i 2i -2i 2i 1-i -1+i 1+i 1+i -1-i -1-i 0 0 1-i -1+i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ22 2 2 -2 -2 0 0 0 0 -2i -2i 2i 2i 1+i -1-i 1-i -1+i 1-i -1+i 0 0 -1-i 1+i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ23 2 2 -2 -2 0 0 0 0 2i 2i -2i -2i 1-i -1+i 1+i -1-i 1+i -1-i 0 0 -1+i 1-i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ24 2 -2 -2 2 0 0 0 0 2i -2i 2i -2i -1-i 1+i -1+i -1+i 1-i 1-i 0 0 -1-i 1+i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ25 2 2 -2 -2 0 0 0 0 -2i -2i 2i 2i -1-i 1+i -1+i 1-i -1+i 1-i 0 0 1+i -1-i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ26 2 -2 -2 2 0 0 0 0 -2i 2i -2i 2i -1+i 1-i -1-i -1-i 1+i 1+i 0 0 -1+i 1-i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ27 2 2 -2 -2 0 0 0 0 2i 2i -2i -2i -1+i 1-i -1-i 1+i -1-i 1+i 0 0 1-i -1+i 0 0 0 0 0 0 complex lifted from C4≀C2 ρ28 2 -2 -2 2 0 0 0 0 2i -2i 2i -2i 1+i -1-i 1-i 1-i -1+i -1+i 0 0 1+i -1-i 0 0 0 0 0 0 complex lifted from C4≀C2

Permutation representations of C2×C4≀C2
On 16 points - transitive group 16T111
Generators in S16
(1 5)(2 6)(3 7)(4 8)(9 15)(10 16)(11 13)(12 14)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 16)(2 15)(3 14)(4 13)(5 10)(6 9)(7 12)(8 11)
(1 5)(2 6)(3 7)(4 8)(9 14 11 16)(10 15 12 13)

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

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

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

G:=TransitiveGroup(16,111);

Matrix representation of C2×C4≀C2 in GL3(𝔽17) generated by

 16 0 0 0 16 0 0 0 16
,
 1 0 0 0 4 0 0 0 13
,
 16 0 0 0 0 13 0 4 0
,
 1 0 0 0 1 0 0 0 4
G:=sub<GL(3,GF(17))| [16,0,0,0,16,0,0,0,16],[1,0,0,0,4,0,0,0,13],[16,0,0,0,0,4,0,13,0],[1,0,0,0,1,0,0,0,4] >;

C2×C4≀C2 in GAP, Magma, Sage, TeX

C_2\times C_4\wr C_2
% in TeX

G:=Group("C2xC4wrC2");
// GroupNames label

G:=SmallGroup(64,101);
// by ID

G=gap.SmallGroup(64,101);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,489,117,88]);
// Polycyclic

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

Export

׿
×
𝔽