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G = C22⋊C4order 16 = 24

The semidirect product of C22 and C4 acting via C4/C2=C2

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

Aliases: C22⋊C4, C2.1D4, C23.C2, C22.2C22, (C2×C4)⋊1C2, C2.1(C2×C4), SmallGroup(16,3)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C22⋊C4
C1C2C22C23 — C22⋊C4
C1C2 — C22⋊C4
C1C22 — C22⋊C4
C1C22 — C22⋊C4

Generators and relations for C22⋊C4
 G = < a,b,c | a2=b2=c4=1, cac-1=ab=ba, bc=cb >

2C2
2C2
2C4
2C22
2C22
2C4

Character table of C22⋊C4

 class 12A2B2C2D2E4A4B4C4D
 size 1111222222
ρ11111111111    trivial
ρ21111-1-1-111-1    linear of order 2
ρ31111-1-11-1-11    linear of order 2
ρ4111111-1-1-1-1    linear of order 2
ρ511-1-1-11i-ii-i    linear of order 4
ρ611-1-11-1-i-iii    linear of order 4
ρ711-1-1-11-ii-ii    linear of order 4
ρ811-1-11-1ii-i-i    linear of order 4
ρ92-2-22000000    orthogonal lifted from D4
ρ102-22-2000000    orthogonal lifted from D4

Permutation representations of C22⋊C4
On 8 points - transitive group 8T10
Generators in S8
(2 6)(4 8)
(1 5)(2 6)(3 7)(4 8)
(1 2 3 4)(5 6 7 8)

G:=sub<Sym(8)| (2,6)(4,8), (1,5)(2,6)(3,7)(4,8), (1,2,3,4)(5,6,7,8)>;

G:=Group( (2,6)(4,8), (1,5)(2,6)(3,7)(4,8), (1,2,3,4)(5,6,7,8) );

G=PermutationGroup([(2,6),(4,8)], [(1,5),(2,6),(3,7),(4,8)], [(1,2,3,4),(5,6,7,8)])

G:=TransitiveGroup(8,10);

Regular action on 16 points - transitive group 16T10
Generators in S16
(1 8)(2 12)(3 6)(4 10)(5 16)(7 14)(9 13)(11 15)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)

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

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

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

G:=TransitiveGroup(16,10);

C22⋊C4 is a maximal subgroup of
C23⋊C4  C42⋊C2  C4×D4  C22≀C2  C4⋊D4  C22.D4  C422C2  A4⋊C4  S32⋊C4  C62⋊C4  D52⋊C4
 C2p.D4: C22⋊Q8  C4.4D4  D6⋊C4  C6.D4  D10⋊C4  C23.D5  D14⋊C4  C23.D7 ...
 Dp.D4, p=1 mod 4: C22⋊F5  D13.D4  D17.D4  D29.D4 ...
C22⋊C4 is a maximal quotient of
C2.C42  C23⋊C4  S32⋊C4  C62⋊C4  D52⋊C4
 D2p⋊C4: D4⋊C4  C4≀C2  D6⋊C4  D10⋊C4  C22⋊F5  D14⋊C4  D22⋊C4  D26⋊C4 ...
 C2p.D4: C22⋊C8  C4.D4  C4.10D4  Q8⋊C4  C6.D4  C23.D5  C23.D7  C23.D11 ...

Polynomial with Galois group C22⋊C4 over ℚ
actionf(x)Disc(f)
8T10x8-13x6+44x4-17x2+1216·34·56·292

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

100
040
031
,
100
040
004
,
200
023
003
G:=sub<GL(3,GF(5))| [1,0,0,0,4,3,0,0,1],[1,0,0,0,4,0,0,0,4],[2,0,0,0,2,0,0,3,3] >;

C22⋊C4 in GAP, Magma, Sage, TeX

C_2^2\rtimes C_4
% in TeX

G:=Group("C2^2:C4");
// GroupNames label

G:=SmallGroup(16,3);
// by ID

G=gap.SmallGroup(16,3);
# by ID

G:=PCGroup([4,-2,2,-2,2,32,49]);
// Polycyclic

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

Export

Subgroup lattice of C22⋊C4 in TeX
Character table of C22⋊C4 in TeX

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