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## G = C2×C8⋊C22order 64 = 26

### Direct product of C2 and C8⋊C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C8⋊C22
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C2×C8⋊C22
 Lower central C1 — C2 — C4 — C2×C8⋊C22
 Upper central C1 — C22 — C22×C4 — C2×C8⋊C22
 Jennings C1 — C2 — C2 — C4 — C2×C8⋊C22

Generators and relations for C2×C8⋊C22
G = < a,b,c,d | a2=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, cd=dc >

Subgroups: 265 in 149 conjugacy classes, 81 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×6], D4 [×11], Q8 [×2], Q8, C23, C23 [×11], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C8⋊C22
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C22×D4, C2×C8⋊C22

Character table of C2×C8⋊C22

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 8A 8B 8C 8D size 1 1 1 1 2 2 4 4 4 4 4 4 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 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 ρ10 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 ρ11 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 ρ12 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 ρ13 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 ρ14 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 ρ15 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 ρ16 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 ρ17 2 2 2 2 2 2 0 0 0 0 0 0 -2 -2 -2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 -2 2 0 0 0 0 0 0 2 -2 2 -2 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 -2 2 -2 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22 ρ22 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C8⋊C22

Permutation representations of C2×C8⋊C22
On 16 points - transitive group 16T89
Generators in S16
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 4)(3 7)(6 8)(10 12)(11 15)(14 16)
(1 5)(3 7)(9 13)(11 15)

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

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

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

G:=TransitiveGroup(16,89);

Matrix representation of C2×C8⋊C22 in GL6(ℤ)

 -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 1 0 0 0 0 0 0 1
,
 -1 -2 0 0 0 0 1 1 0 0 0 0 0 0 1 0 2 0 0 0 1 0 1 1 0 0 0 -1 -1 0 0 0 0 0 -1 0
,
 -1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 1 -1 0 0 0 0 -1 0 0 -1 0 0 -1 0 -1 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 -1 0 -1 0 0 0 -1 0 0 -1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,-1,0,0,0,2,1,-1,-1,0,0,0,1,0,0],[-1,1,0,0,0,0,0,1,0,0,0,0,0,0,1,1,-1,-1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,-1,-1,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;

C2×C8⋊C22 in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes C_2^2
% in TeX

G:=Group("C2xC8:C2^2");
// GroupNames label

G:=SmallGroup(64,254);
// by ID

G=gap.SmallGroup(64,254);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,-2,217,650,1444,730,88]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,c*d=d*c>;
// generators/relations

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