Finite Group Databasebeta

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C₂×C₄×D₇×AGL(1,𝔽₅) #72217754

order	2240
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂×C₄×D₇×AGL(1,𝔽₅)

Splittings

Fitting	C₂×C₄×C₅×C₇ ⋊ C₂×C₄
derived, Hall	C₅×C₇ ⋊ C₂²×C₄²
Sylow	C₇ ⋊ C₂²×C₄×AGL(1,𝔽₅)
Sylow	C₅ ⋊ C₂×C₄²×D₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	D₇×AGL(1,𝔽₅)
commutator, hypocenter	C₅×C₇	C₂²×C₄²
Frattini	C₂	C₂²×D₇×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×C₇	C₂×C₄
socle	C₂²×C₅×C₇	C₂²×C₄
prime radical	C₂²×D₅×D₇	C₂²
cosocle	C₂×C₇×D₅	C₂⁴
LCS	G ▷ C₅×C₇
UCS	C₁ ◁ C₂×C₄

996 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₄×AGL(1,𝔽₅)×AGL(1,𝔽₇), C₂×D₄×D₇×AGL(1,𝔽₅), D₇×M₁₆×AGL(1,𝔽₅), …
maximal normal subgroup of	C₂×C₄×AGL(1,𝔽₅)×AGL(1,𝔽₇), C₂×D₄×D₇×AGL(1,𝔽₅), D₇×M₁₆×AGL(1,𝔽₅), …
maximal quotient of	D₇×M₁₆×AGL(1,𝔽₅), D₇×C₂²⋊C₄×AGL(1,𝔽₅)

GAP

Past the SmallGroups library coverage — no SmallGroups id.