Finite Group Databasebeta

← Prev (#117) Next (#119) →

C₄×D₄ #118

order	32
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×#120 (order 128) · Inn(G) = C₂² · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₄×D₄

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²
commutator	C₂	C₂²×C₄
Frattini, socle, cosocle	C₂²	C₂³
prime radical	C₂×D₄	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

47 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 8 points:

(1,2,3,4);(5,6)(7,8);(6,8)

Conjugacy classes & character table

20 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₄×D₄×A₅, C₄×#91881, D₄×#6732, …
maximal subgroup of	D₄⋊C₈, C₈×D₄, #435, …
maximal normal subgroup of	D₄⋊C₈, C₈×D₄, #435, …
maximal quotient of	C₄×C₂²⋊C₄, C₄×C₄⋊C₄, #385, …
automorphism group of	C₂×C₄×C₅, C₅×D₄
outer automorphism group of	C₂×C₄×C₅, #366, #367, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32, 25);