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AGL(1,𝔽₇)×C₃²⋊C₉ #72249143

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₉ · Schur multiplier (H₂) = C₃⁴

Aliases

AGL(1,𝔽₇)×C₃²⋊C₉

Splittings

Fitting	C₇×C₃²⋊C₉ ⋊ C₂×C₃
Hall	ΓL(1,𝔽₈)×C₃²⋊C₉ ⋊ C₂
Hall	D₇ ⋊ C₃×C₃²⋊C₉
Sylow	C₇ ⋊ C₂×C₃×C₃²⋊C₉

Subgroups

	subgroup	G/H
center, Frattini	C₃²	C₃²×AGL(1,𝔽₇)
commutator	C₃×C₇	C₂×C₃²×C₉
Fitting	C₇×C₃²⋊C₉	C₂×C₃
socle, cosocle	C₃²×C₇	C₂×C₃³
prime radical	C₃³×AGL(1,𝔽₇)	C₃
hypercenter	C₃²⋊C₉	AGL(1,𝔽₇)
hypocenter	C₇	C₂×C₃×C₃²⋊C₉
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

576 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 189

Conjugacy classes & character table

231 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#6891, AGL(1,𝔽₇)×#4181
maximal normal subgroup of	AGL(1,𝔽₇)×#6891, AGL(1,𝔽₇)×#4181

GAP

Past the SmallGroups library coverage — no SmallGroups id.