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## G = C2×C4⋊C4order 32 = 25

### Direct product of C2 and C4⋊C4

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

Aliases: C2×C4⋊C4, C22.3Q8, C22.13D4, C22.5C23, C23.13C22, C42(C2×C4), (C2×C4)⋊3C4, C2.2(C2×D4), C2.1(C2×Q8), C2.2(C22×C4), (C2×C4).9C22, (C22×C4).3C2, C22.10(C2×C4), SmallGroup(32,23)

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×C4⋊C4
G = < a,b,c | a2=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×C4⋊C4

 class 1 2A 2B 2C 2D 2E 2F 2G 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 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 trivial ρ2 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 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 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 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 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 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 linear of order 2 ρ9 1 -1 -1 1 -1 1 -1 1 -1 1 -i i i -i 1 -1 i -i -i i linear of order 4 ρ10 1 -1 -1 1 1 -1 1 -1 -1 -1 i i -i -i 1 1 -i -i i i linear of order 4 ρ11 1 -1 -1 1 1 -1 1 -1 1 1 -i -i -i -i -1 -1 i i i i linear of order 4 ρ12 1 -1 -1 1 -1 1 -1 1 1 -1 i -i i -i -1 1 -i i -i i linear of order 4 ρ13 1 -1 -1 1 -1 1 -1 1 1 -1 -i i -i i -1 1 i -i i -i linear of order 4 ρ14 1 -1 -1 1 1 -1 1 -1 1 1 i i i i -1 -1 -i -i -i -i linear of order 4 ρ15 1 -1 -1 1 1 -1 1 -1 -1 -1 -i -i i i 1 1 i i -i -i linear of order 4 ρ16 1 -1 -1 1 -1 1 -1 1 -1 1 i -i -i i 1 -1 -i i i -i linear of order 4 ρ17 2 -2 2 -2 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 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 0 0 0 symplectic lifted from Q8, Schur index 2 ρ20 2 2 -2 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 symplectic lifted from Q8, Schur index 2

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

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

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

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

Matrix representation of C2×C4⋊C4 in GL4(𝔽5) generated by

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

C2×C4⋊C4 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes C_4
% in TeX

G:=Group("C2xC4:C4");
// GroupNames label

G:=SmallGroup(32,23);
// by ID

G=gap.SmallGroup(32,23);
# by ID

G:=PCGroup([5,-2,2,2,-2,2,80,101,46]);
// Polycyclic

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

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