C16:C2^2 - GroupNames

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G = C₁₆⋊C₂² order 64 = 2⁶

The semidirect product of C₁₆ and C₂² acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C₁₆⋊C₂², C₈.₃D₄, D₁₆⋊₂C₂, C₄.₁₄D₈, SD₃₂⋊₁C₂, D₈⋊₂C₂², C₂².₅D₈, M₅(2)⋊₁C₂, C₈.₁₀C₂³, Q₁₆⋊₂C₂², C₄○D₈⋊₂C₂, (C₂×D₈)⋊₁₀C₂, (C₂×C₄).₄₇D₄, C₄.₁₁(C₂×D₄), C₂.₁₆(C₂×D₈), (C₂×C₈).₂₃C₂², 2-Sylow(PGammaL(2,49)), SmallGroup(64,190)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

C₁ — C₈ — C₁₆⋊C₂²

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂×D₈ — C₁₆⋊C₂²

C₁ — C₂ — C₄ — C₈ — C₁₆⋊C₂²

C₁ — C₂ — C₂×C₄ — C₂×C₈ — C₁₆⋊C₂²

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₈ — C₁₆⋊C₂²

Generators and relations for C₁₆⋊C₂²
G = < a,b,c | a¹⁶=b²=c²=1, bab=a⁷, cac=a⁹, bc=cb >

C₁

C₂

₂C₂

₈C₂

₈C₂

₈C₂

C₄

C₂²

C₄

₄C₄

₄C₂²

₄C₂²

₄C₂²

₈C₂²

₈C₂²

C₈

C₈

₂D₄

₂D₄

₂D₄

₂Q₈

₄D₄

₄C₂³

₄D₄

₄C₂×C₄

C₁₆

C₁₆

D₈

D₈

Q₁₆

D₈

₂C₄○D₄

₂SD₁₆

₂D₈

₂C₂×D₄

SD₃₂

D₁₆

SD₃₂

D₁₆

C₁₆⋊C₂²

Character table of C₁₆⋊C₂²

class	1	2A	2B	2C	2D	2E	4A	4B	4C	8A	8B	8C	16A	16B	16C	16D
size	1	1	2	8	8	8	2	2	8	2	2	4	4	4	4	4
ρ₁	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	linear of order 2
ρ₃	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	linear of order 2
ρ₅	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	linear of order 2
ρ₇	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	linear of order 2
ρ₉	2	2	2	0	0	0	2	2	0	-2	-2	-2	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	0	0	0	2	-2	0	-2	-2	2	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	0	0	0	-2	2	0	0	0	0	-√2	√2	√2	-√2	orthogonal lifted from D₈
ρ₁₂	2	2	-2	0	0	0	-2	2	0	0	0	0	√2	-√2	-√2	√2	orthogonal lifted from D₈
ρ₁₃	2	2	2	0	0	0	-2	-2	0	0	0	0	√2	√2	-√2	-√2	orthogonal lifted from D₈
ρ₁₄	2	2	2	0	0	0	-2	-2	0	0	0	0	-√2	-√2	√2	√2	orthogonal lifted from D₈
ρ₁₅	4	-4	0	0	0	0	0	0	0	-2√2	2√2	0	0	0	0	0	orthogonal faithful
ρ₁₆	4	-4	0	0	0	0	0	0	0	2√2	-2√2	0	0	0	0	0	orthogonal faithful

Permutation representations of C₁₆⋊C₂²
►On 16 points - transitive group 16T134

Generators in S₁₆

(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)

(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)

(1 9)(3 11)(5 13)(7 15)

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

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

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

G:=TransitiveGroup(16,134);

C₁₆⋊C₂² is a maximal subgroup of
D₁₆⋊C₄
D₈⋊D_2p: D₈⋊D₄ D₈⋊₃D₄ D₈⋊D₆ Q₁₆⋊D₆ D₁₆⋊D₅ D₈⋊D₁₀ D₈⋊D₁₄ Q₁₆⋊D₁₄ ...
D₈.D_2p: Q₁₆.₁₀D₄ Q₁₆.D₄ C₈.₃D₈ D₄₈⋊C₂ D₈.D₆ C₁₆⋊D₁₀ D₈.D₁₀ D₁₁₂⋊C₂ ...
D_8p⋊C₂²: D₁₆⋊C₂² D₄○D₁₆ D₄○SD₃₂ C₁₆⋊D₆ D₈₀⋊C₂ C₁₆⋊D₁₄ ...
C₁₆⋊C₂² is a maximal quotient of
C₂³.₃₉D₈ C₂³.₄₀D₈ M₅(2)⋊₁C₄ SD₃₂⋊₃C₄ D₁₆⋊₄C₄ D₈⋊₇D₄ D₈⋊₂D₄ D₈⋊Q₈ C₄.Q₃₂ D₈.Q₈ C₂³.₄₉D₈ C₂³.₁₉D₈ C₂³.₅₁D₈ C₈.₁₂SD₁₆ C₈.₁₃SD₁₆ C₁₆⋊Q₈
C₁₆⋊D_2p: C₁₆⋊D₄ C₁₆⋊₂D₄ C₁₆⋊₃D₄ C₁₆⋊D₆ D₈⋊D₆ D₄₈⋊C₂ D₈₀⋊C₂ D₁₆⋊D₅ ...
D₈.D_2p: D₈.₉D₄ D₈.₅D₄ D₈.D₆ D₈.D₁₀ D₈.D₁₄ ...
Q₁₆⋊D_2p: D₈⋊₈D₄ Q₁₆⋊₂D₄ Q₁₆⋊D₆ D₈⋊D₁₀ Q₁₆⋊D₁₄ ...

Matrix representation of C₁₆⋊C₂² ►in GL₄(𝔽₇) generated by

0	5	3	6
5	2	5	0
1	3	1	4
5	5	2	4

,

0	4	0	1
4	1	0	4
4	4	1	6
6	3	0	5

,

1	0	6	3
0	2	3	2
0	3	2	2
0	1	1	2

G:=sub<GL(4,GF(7))| [0,5,1,5,5,2,3,5,3,5,1,2,6,0,4,4],[0,4,4,6,4,1,4,3,0,0,1,0,1,4,6,5],[1,0,0,0,0,2,3,1,6,3,2,1,3,2,2,2] >;

C₁₆⋊C₂² in GAP, Magma, Sage, TeX

C_{16}\rtimes C_2^2

% in TeX

G:=Group("C16:C2^2");

// GroupNames label

G:=SmallGroup(64,190);

// by ID

G=gap.SmallGroup(64,190);

# by ID

G:=PCGroup([6,-2,2,2,-2,-2,-2,121,650,579,297,165,1444,730,88]);

// Polycyclic

G:=Group<a,b,c|a^16=b^2=c^2=1,b*a*b=a^7,c*a*c=a^9,b*c=c*b>;

// generators/relations

Export

Subgroup lattice of C₁₆⋊C₂² in TeX
Character table of C₁₆⋊C₂² in TeX

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