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C₃×(C₁₂₇⋊C₃)⋊D₇ #30630713

order	16002
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃×(C₁₂₇⋊C₃)⋊D₇
unambiguous semidirect product	(C₁₂₇⋊C₃)⋊(C₃×D₇)

Splittings

Fitting, socle	C₃×C₇×C₁₂₇ ⋊ C₂×C₃
derived, Hall	C₇×C₁₂₇ ⋊ C₂×C₃²
Hall	C₃×C₇×C₁₂₇⋊C₃ ⋊ C₂
Hall	D₈₈₉ ⋊ C₃²
Hall	C₃×C₁₂₇⋊C₃ ⋊ D₇
Sylow	C₁₂₇ ⋊ C₃²×D₇
Sylow	C₇ ⋊ C₃×D₁₂₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	(C₁₂₇⋊C₃)⋊D₇
commutator, cosocle, hypocenter	C₇×C₁₂₇	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇×C₁₂₇	C₂×C₃
LCS	G ▷ C₇×C₁₂₇
UCS	C₁ ◁ C₃

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

486 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×D₇×D₁₂₇⋊C₃, D₃×(C₁₂₇⋊C₃)⋊D₇, (C₃⋊(D₁₂₇⋊C₃))⋊D₇, …
maximal normal subgroup of	C₃×D₇×D₁₂₇⋊C₃, D₃×(C₁₂₇⋊C₃)⋊D₇, (C₃⋊(D₁₂₇⋊C₃))⋊D₇, …
maximal quotient of	C₃×(C₁₂₇⋊C₃)⋊Dic₇, C₂×C₃×(C₁₂₇⋊C₃)⋊D₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16002, 70);