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## G = C2×C4⋊D4order 64 = 26

### Direct product of C2 and C4⋊D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C4⋊D4
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 353 in 213 conjugacy classes, 97 normal (13 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, C22×D4, C2×C4⋊D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, C2×C4⋊D4

Character table of C2×C4⋊D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L size 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 2 2 2 2 2 2 2 2 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 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 -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 -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 -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 -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 -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 -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 -1 -1 -1 -1 1 1 linear of order 2 ρ17 2 2 2 -2 -2 2 -2 -2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 -2 -2 -2 -2 2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 2 0 0 -2 -2 0 2 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 -2 0 0 2 2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ21 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 2 0 0 2 -2 0 -2 0 0 0 0 orthogonal lifted from D4 ρ22 2 -2 -2 2 -2 2 -2 2 0 0 0 0 0 0 0 0 0 -2 0 0 -2 2 0 2 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 -2 -2 2 2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 2 -2 -2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 2i 0 -2i -2i 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 -2i 0 2i 2i 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 -2 -2 2 2 2 -2 0 0 0 0 0 0 0 0 -2i 0 -2i 2i 0 0 2i 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 -2 -2 -2 2 2 2 -2 0 0 0 0 0 0 0 0 2i 0 2i -2i 0 0 -2i 0 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C2×C4⋊D4
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 12)(6 9)(7 10)(8 11)(13 31)(14 32)(15 29)(16 30)(17 22)(18 23)(19 24)(20 21)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 15 11 22)(2 14 12 21)(3 13 9 24)(4 16 10 23)(5 20 26 32)(6 19 27 31)(7 18 28 30)(8 17 25 29)
(1 6)(2 5)(3 8)(4 7)(9 25)(10 28)(11 27)(12 26)(13 29)(14 32)(15 31)(16 30)(17 24)(18 23)(19 22)(20 21)

G:=sub<Sym(32)| (1,25)(2,26)(3,27)(4,28)(5,12)(6,9)(7,10)(8,11)(13,31)(14,32)(15,29)(16,30)(17,22)(18,23)(19,24)(20,21), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,15,11,22)(2,14,12,21)(3,13,9,24)(4,16,10,23)(5,20,26,32)(6,19,27,31)(7,18,28,30)(8,17,25,29), (1,6)(2,5)(3,8)(4,7)(9,25)(10,28)(11,27)(12,26)(13,29)(14,32)(15,31)(16,30)(17,24)(18,23)(19,22)(20,21)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,12)(6,9)(7,10)(8,11)(13,31)(14,32)(15,29)(16,30)(17,22)(18,23)(19,24)(20,21), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,15,11,22)(2,14,12,21)(3,13,9,24)(4,16,10,23)(5,20,26,32)(6,19,27,31)(7,18,28,30)(8,17,25,29), (1,6)(2,5)(3,8)(4,7)(9,25)(10,28)(11,27)(12,26)(13,29)(14,32)(15,31)(16,30)(17,24)(18,23)(19,22)(20,21) );

G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,12),(6,9),(7,10),(8,11),(13,31),(14,32),(15,29),(16,30),(17,22),(18,23),(19,24),(20,21)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,15,11,22),(2,14,12,21),(3,13,9,24),(4,16,10,23),(5,20,26,32),(6,19,27,31),(7,18,28,30),(8,17,25,29)], [(1,6),(2,5),(3,8),(4,7),(9,25),(10,28),(11,27),(12,26),(13,29),(14,32),(15,31),(16,30),(17,24),(18,23),(19,22),(20,21)]])

Matrix representation of C2×C4⋊D4 in GL5(𝔽5)

 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 3 1 0 0 0 0 2 0 0 0 0 0 1 3 0 0 0 1 4
,
 1 0 0 0 0 0 1 2 0 0 0 4 4 0 0 0 0 0 4 0 0 0 0 4 1
,
 1 0 0 0 0 0 1 0 0 0 0 4 4 0 0 0 0 0 1 0 0 0 0 1 4

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,3,0,0,0,0,1,2,0,0,0,0,0,1,1,0,0,0,3,4],[1,0,0,0,0,0,1,4,0,0,0,2,4,0,0,0,0,0,4,4,0,0,0,0,1],[1,0,0,0,0,0,1,4,0,0,0,0,4,0,0,0,0,0,1,1,0,0,0,0,4] >;

C2×C4⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes D_4
% in TeX

G:=Group("C2xC4:D4");
// GroupNames label

G:=SmallGroup(64,203);
// by ID

G=gap.SmallGroup(64,203);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650]);
// Polycyclic

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

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