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C₄²×D₃×ΓL(1,𝔽₈) #72244247

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄²×D₃×ΓL(1,𝔽₈)

Splittings

Fitting	C₃×C₄²×C₇ ⋊ C₂×C₃
derived	C₃×C₇ ⋊ C₂×C₃×C₄²
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×C₄²
Sylow	C₇ ⋊ C₃×C₄²×D₃

Subgroups

	subgroup	G/H
center, hypercenter	C₄²	D₃×ΓL(1,𝔽₈)
commutator, hypocenter	C₃×C₇	C₂×C₃×C₄²
Frattini	C₂²	C₂²×D₃×ΓL(1,𝔽₈)
Fitting	C₃×C₄²×C₇	C₂×C₃
socle, cosocle	C₂²×C₃×C₇	C₂³×C₃
prime radical	C₂²×D₃×ΓL(1,𝔽₈)	C₂²
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₄²

462 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 207 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 336

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×ΓL(1,𝔽₈)×#127, C₄²×D₃×AGL(1,𝔽₇), D₃×#10620710, …
maximal normal subgroup of	D₃×ΓL(1,𝔽₈)×#127, C₄²×D₃×AGL(1,𝔽₇), D₃×#10620710, …
maximal quotient of	C₄²×C₃⋊D₃×ΓL(1,𝔽₈)

GAP

Past the SmallGroups library coverage — no SmallGroups id.