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D₇×#5241 #72244284

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂²
Hall	C₃×C₇ ⋊ C₂×#453
Sylow	C₇ ⋊ C₂×#5241
Sylow	C₃ ⋊ D₇×#453

Subgroups

	subgroup	G/H
center	C₂	D₇×#953
commutator, cosocle	C₂×C₃×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₇
Fitting	C₃×C₇×H(5,𝔽₂)	C₂²
socle	C₂×C₃×C₇	C₂×#120
hypercenter	H(5,𝔽₂)	D₃×D₇
hypocenter	C₃×C₇	C₂×#453
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

1688 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Conjugacy classes & character table

165 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₇×#453, D₇×#10591510, AGL(1,𝔽₇)×#5241
maximal normal subgroup of	D₃×D₇×#453, AGL(1,𝔽₇)×#5241
maximal quotient of	D₇×#10591510

GAP

Past the SmallGroups library coverage — no SmallGroups id.