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C₃×C₄²×AGL(1,𝔽₇) #72244233

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₄² · Schur multiplier (H₂) = C₂²×C₃×C₄

Aliases

C₃×C₄²×AGL(1,𝔽₇)

Splittings

Fitting	C₃×C₄²×C₇ ⋊ C₂×C₃
derived, Sylow	C₇ ⋊ C₂×C₃²×C₄²
Hall	C₄²×D₇ ⋊ C₃²
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄²	AGL(1,𝔽₇)
commutator, hypocenter	C₇	C₂×C₃²×C₄²
Frattini	C₂²	C₂²×C₃×AGL(1,𝔽₇)
Fitting	C₃×C₄²×C₇	C₂×C₃
socle	C₂²×C₃×C₇	C₂³×C₃
prime radical	C₂²×C₃×AGL(1,𝔽₇)	C₂²
cosocle	C₂²×C₇	C₂³×C₃²
LCS	G ▷ C₇
UCS	C₁ ◁ C₃×C₄²

648 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 354 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

336 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×#127×AGL(1,𝔽₇), C₄²×D₃×AGL(1,𝔽₇), AGL(1,𝔽₇)×#874, …
maximal normal subgroup of	C₃×#127×AGL(1,𝔽₇), C₄²×D₃×AGL(1,𝔽₇), AGL(1,𝔽₇)×#874, …
maximal quotient of	C₄²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), C₃²×C₄²×AGL(1,𝔽₇)

GAP

Past the SmallGroups library coverage — no SmallGroups id.