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## G = C2×D4⋊C4order 64 = 26

### Direct product of C2 and D4⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×D4⋊C4
G = < a,b,c,d | a2=b4=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=dbd-1=b-1, dcd-1=bc >

Subgroups: 201 in 101 conjugacy classes, 49 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×4], D4 [×6], C23, C23 [×10], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C24, D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C2×D4⋊C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C2×D4⋊C4

Character table of C2×D4⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 1 1 1 1 1 1 4 4 4 4 2 2 2 2 4 4 4 4 2 2 2 2 2 2 2 2 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 -i -i i i i -i i -i -i i -i i linear of order 4 ρ10 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 1 1 i -i -i i -i -i -i -i i i i i linear of order 4 ρ11 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 1 1 -i i i -i i i i i -i -i -i -i linear of order 4 ρ12 1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 i i -i -i -i i -i i i -i i -i linear of order 4 ρ13 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 -1 1 1 -i i i -i -i -i -i -i i i i i linear of order 4 ρ14 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 i i -i -i i -i i -i -i i -i i linear of order 4 ρ15 1 1 -1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -i -i i i -i i -i i i -i i -i linear of order 4 ρ16 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 -1 1 1 i -i -i i i i i i -i -i -i -i linear of order 4 ρ17 2 2 -2 2 -2 2 -2 -2 0 0 0 0 2 -2 2 -2 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 0 0 0 0 -2 2 2 -2 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 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 -2 2 2 2 -2 0 0 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 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 0 0 0 -√2 -√2 √2 √2 -√2 -√2 √2 √2 orthogonal lifted from D8 ρ22 2 2 2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -√2 √2 √2 -√2 -√2 √2 √2 -√2 orthogonal lifted from D8 ρ23 2 2 2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 -√2 -√2 √2 √2 -√2 -√2 √2 orthogonal lifted from D8 ρ24 2 -2 2 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 √2 √2 -√2 -√2 √2 √2 -√2 -√2 orthogonal lifted from D8 ρ25 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 √-2 √-2 -√-2 -√-2 -√-2 -√-2 √-2 √-2 complex lifted from SD16 ρ26 2 -2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 √-2 √-2 -√-2 √-2 -√-2 -√-2 √-2 complex lifted from SD16 ρ27 2 -2 -2 2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 √-2 -√-2 -√-2 √-2 -√-2 √-2 √-2 -√-2 complex lifted from SD16 ρ28 2 2 -2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -√-2 -√-2 √-2 √-2 √-2 √-2 -√-2 -√-2 complex lifted from SD16

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

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

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

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

Matrix representation of C2×D4⋊C4 in GL4(𝔽17) generated by

 16 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
,
 16 0 0 0 0 16 0 0 0 0 0 1 0 0 1 0
,
 4 0 0 0 0 13 0 0 0 0 12 5 0 0 5 5
G:=sub<GL(4,GF(17))| [16,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0],[4,0,0,0,0,13,0,0,0,0,12,5,0,0,5,5] >;

C2×D4⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes C_4
% in TeX

G:=Group("C2xD4:C4");
// GroupNames label

G:=SmallGroup(64,95);
// by ID

G=gap.SmallGroup(64,95);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,-2,96,121,963,489,117]);
// 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=d*b*d^-1=b^-1,d*c*d^-1=b*c>;
// generators/relations

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