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C₂²×C₁₅₇⋊C₁₃ #22551023

order	8164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂²×C₁₅₇⋊C₁₃
unambiguous semidirect product	(C₂²×C₁₅₇)⋊C₁₃

Splittings

Fitting, socle, Hall	C₂²×C₁₅₇ ⋊ C₁₃
derived, Sylow	C₁₅₇ ⋊ C₂²×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	C₁₅₇⋊C₁₃
commutator, hypocenter	C₁₅₇	C₂²×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₁₅₇	C₁₃
LCS	G ▷ C₁₅₇
UCS	C₁ ◁ C₂²

20 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊((C₁₅₇⋊C₃)⋊C₁₃), A₄×C₁₅₇⋊C₁₃, C₂²×(C₁₅₇⋊C₃)⋊C₁₃, …
maximal normal subgroup of	C₂²⋊((C₁₅₇⋊C₃)⋊C₁₃), A₄×C₁₅₇⋊C₁₃, C₂²×(C₁₅₇⋊C₃)⋊C₁₃, …
maximal quotient of	C₂×D₃×C₁₅₇⋊C₁₃, C₂²×C₃×C₁₅₇⋊C₁₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8164, 10);