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C₄×D₃×H(5,𝔽₂) #11701786

order	768
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

C₄×D₃×H(5,𝔽₂)

Splittings

Fitting	C₃×C₄×H(5,𝔽₂) ⋊ C₂
Sylow	C₃ ⋊ C₂×C₄×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂⁴×D₃
commutator	C₂×C₃	C₂⁵×C₄
Frattini	C₂²	C₂⁵×D₃
Fitting	C₃×C₄×H(5,𝔽₂)	C₂
socle, cosocle	C₂²×C₃	C₂⁶
prime radical	C₂×D₃×H(5,𝔽₂)	C₂
hypercenter	C₄×H(5,𝔽₂)	D₃
hypocenter	C₃	C₂×C₄×H(5,𝔽₂)
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×H(5,𝔽₂)

12378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 96

Conjugacy classes & character table

204 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#60176, C₄×#10596675, D₃×AGL(1,𝔽₅)×H(5,𝔽₂), …
maximal normal subgroup of	D₃×#60176, C₄×D₃×#453, C₄×D₃×B(4,𝔽₂), …
maximal quotient of	C₄×#10596675, D₃×AGL(1,𝔽₅)×H(5,𝔽₂), D₃×#34415, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1076167);