Finite Group Databasebeta

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D₇×#6911 #72249104

order	3402
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³ · Schur multiplier (H₂) = C₃⁴

Aliases

Splittings

Fitting, Hall	C₇×#6911 ⋊ C₂
Sylow	C₇ ⋊ C₂×#6911

Subgroups

	subgroup	G/H
center	C₃	C₃×D₇×H(3,𝔽₃)
commutator, cosocle	C₃²×C₇	C₂×C₃³
Frattini	C₃²	C₃³×D₇
Fitting	C₇×#6911	C₂
socle	C₃×C₇	C₂×C₃×H(3,𝔽₃)
hypercenter	#6911	D₇
hypocenter	C₇	C₂×#6911
LCS	G ▷ C₃²×C₇ ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 189

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#10624749, AGL(1,𝔽₇)×#6911, #72249154, …
maximal normal subgroup of	D₇×#10624749, AGL(1,𝔽₇)×#6911, #72249154, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.