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## G = C2×C42⋊2C2order 64 = 26

### Direct product of C2 and C42⋊2C2

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

Aliases: C2×C422C2, C4214C22, C22.22C24, C23.72C23, C24.14C22, (C2×C42)⋊4C2, C4⋊C413C22, (C2×C4).53C23, C22.34(C4○D4), C22⋊C4.12C22, (C22×C4).60C22, (C2×C4⋊C4)⋊17C2, C2.11(C2×C4○D4), (C2×C22⋊C4).12C2, SmallGroup(64,209)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C42⋊2C2
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊2C2
 Lower central C1 — C22 — C2×C42⋊2C2
 Upper central C1 — C23 — C2×C42⋊2C2
 Jennings C1 — C22 — C2×C42⋊2C2

Generators and relations for C2×C422C2
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 177 in 123 conjugacy classes, 81 normal (6 characteristic)
C1, C2 [×7], C2 [×2], C4 [×12], C22, C22 [×6], C22 [×10], C2×C4 [×12], C2×C4 [×12], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C24, C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C422C2 [×8], C2×C422C2
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C422C2 [×4], C2×C4○D4 [×3], C2×C422C2

Character table of C2×C422C2

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R size 1 1 1 1 1 1 1 1 4 4 2 2 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 -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 2i 0 2i -2i 0 0 0 0 0 0 -2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 -2 2 -2 -2 2 0 0 2i 0 -2i -2i 0 0 0 0 0 0 2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ19 2 2 -2 2 -2 2 -2 -2 0 0 0 -2i 0 0 0 0 0 0 -2i 2i 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 -2i -2i 2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ21 2 -2 2 -2 -2 2 -2 2 0 0 0 0 0 0 2i 2i -2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 -2 -2 2 -2 -2 2 2 0 0 0 2i 0 0 0 0 0 0 -2i -2i 0 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 2 2 -2 -2 2 -2 -2 2 0 0 -2i 0 2i 2i 0 0 0 0 0 0 -2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ24 2 -2 -2 -2 2 2 2 -2 0 0 -2i 0 -2i 2i 0 0 0 0 0 0 2i 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ25 2 2 -2 2 -2 2 -2 -2 0 0 0 2i 0 0 0 0 0 0 2i -2i 0 -2i 0 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 -2i 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 -2 -2 2 -2 -2 2 2 0 0 0 -2i 0 0 0 0 0 0 2i 2i 0 -2i 0 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 2i -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4

Smallest permutation representation of C2×C422C2
On 32 points
Generators in S32
(1 25)(2 26)(3 27)(4 28)(5 16)(6 13)(7 14)(8 15)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 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 17 13 31)(2 18 14 32)(3 19 15 29)(4 20 16 30)(5 24 28 10)(6 21 25 11)(7 22 26 12)(8 23 27 9)
(1 27)(2 5)(3 25)(4 7)(6 15)(8 13)(9 29)(10 20)(11 31)(12 18)(14 28)(16 26)(17 21)(19 23)(22 32)(24 30)

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

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

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

Matrix representation of C2×C422C2 in GL5(𝔽5)

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

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

C2×C422C2 in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C2xC4^2:2C2");
// GroupNames label

G:=SmallGroup(64,209);
// by ID

G=gap.SmallGroup(64,209);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,86]);
// 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,b*c=c*b,d*b*d=b*c^2,d*c*d=b^2*c^-1>;
// generators/relations

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