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## G = C2×C4.4D4order 64 = 26

### Direct product of C2 and C4.4D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C4.4D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4.4D4
 Lower central C1 — C22 — C2×C4.4D4
 Upper central C1 — C23 — C2×C4.4D4
 Jennings C1 — C22 — C2×C4.4D4

Generators and relations for C2×C4.4D4
G = < a,b,c,d | a2=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 265 in 165 conjugacy classes, 89 normal (9 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C2×C4.4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C22×D4, C2×C4○D4, C2×C4.4D4

Character table of C2×C4.4D4

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

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

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

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

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

Matrix representation of C2×C4.4D4 in GL5(𝔽5)

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

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

C2×C4.4D4 in GAP, Magma, Sage, TeX

C_2\times C_4._4D_4
% in TeX

G:=Group("C2xC4.4D4");
// GroupNames label

G:=SmallGroup(64,207);
// by ID

G=gap.SmallGroup(64,207);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,199,650,86]);
// Polycyclic

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

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