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G = C22×C4order 16 = 24

Abelian group of type [2,2,4]

direct product, p-group, abelian, monomial

Aliases: C22×C4, SmallGroup(16,10)

Series: Derived Chief Lower central Upper central Jennings

C1 — C22×C4
C1C2C22C23 — C22×C4
C1 — C22×C4
C1 — C22×C4
C1C2 — C22×C4

Generators and relations for C22×C4
 G = < a,b,c | a2=b2=c4=1, ab=ba, ac=ca, bc=cb >


Character table of C22×C4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H
 size 1111111111111111
ρ11111111111111111    trivial
ρ211-1-111-1-111-1-111-1-1    linear of order 2
ρ3111-1-1-11-11-11-11-11-1    linear of order 2
ρ411-11-1-1-111-1-111-1-11    linear of order 2
ρ511111111-1-1-1-1-1-1-1-1    linear of order 2
ρ611-1-111-1-1-1-111-1-111    linear of order 2
ρ7111-1-1-11-1-11-11-11-11    linear of order 2
ρ811-11-1-1-11-111-1-111-1    linear of order 2
ρ91-1111-1-1-1iiii-i-i-i-i    linear of order 4
ρ101-1-1-11-111ii-i-i-i-iii    linear of order 4
ρ111-11-1-11-11i-ii-i-ii-ii    linear of order 4
ρ121-1-11-111-1i-i-ii-iii-i    linear of order 4
ρ131-1111-1-1-1-i-i-i-iiiii    linear of order 4
ρ141-1-1-11-111-i-iiiii-i-i    linear of order 4
ρ151-11-1-11-11-ii-iii-ii-i    linear of order 4
ρ161-1-11-111-1-iii-ii-i-ii    linear of order 4

Permutation representations of C22×C4
Regular action on 16 points - transitive group 16T2
Generators in S16
(1 8)(2 5)(3 6)(4 7)(9 15)(10 16)(11 13)(12 14)
(1 15)(2 16)(3 13)(4 14)(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

G:=sub<Sym(16)| (1,8)(2,5)(3,6)(4,7)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)>;

G:=Group( (1,8)(2,5)(3,6)(4,7)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16) );

G=PermutationGroup([(1,8),(2,5),(3,6),(4,7),(9,15),(10,16),(11,13),(12,14)], [(1,15),(2,16),(3,13),(4,14),(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)])

G:=TransitiveGroup(16,2);

Polynomial with Galois group C22×C4 over ℚ
actionf(x)Disc(f)
16T2x16-x12+x8-x4+1232·512

Matrix representation of C22×C4 in GL3(𝔽5) generated by

100
040
004
,
400
010
001
,
100
040
002
G:=sub<GL(3,GF(5))| [1,0,0,0,4,0,0,0,4],[4,0,0,0,1,0,0,0,1],[1,0,0,0,4,0,0,0,2] >;

C22×C4 in GAP, Magma, Sage, TeX

C_2^2\times C_4
% in TeX

G:=Group("C2^2xC4");
// GroupNames label

G:=SmallGroup(16,10);
// by ID

G=gap.SmallGroup(16,10);
# by ID

G:=PCGroup([4,-2,2,2,-2,32]);
// Polycyclic

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

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