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C₈×D₄ #434

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂³×#120 (order 256) · Inn(G) = C₂² · Out(G) = C₂³×D₄
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

C₈×D₄

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²
commutator	C₂	C₂²×C₈
Frattini, cosocle	C₂×C₄	C₂³
socle	C₂²	C₂²×C₄
prime radical	C₂×D₄	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 12 points:

(5,6,7,9,8,10,11,12);(1,2)(3,4);(2,4)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄⋊C₁₆, C₈×D₈, C₈×SD₁₆, …
maximal normal subgroup of	D₄⋊C₁₆, C₈×D₈, C₈×SD₁₆, …
maximal quotient of	C₈×D₈, C₈×SD₁₆, C₈×Q₁₆, …
outer automorphism group of	#1423, #1424, C₂×Dic₁₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 115);