Finite Group Databasebeta

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C₂²×#14522510 #72275673

order	8000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂²×#14522510

Splittings

Fitting	C₂²×H(3,𝔽₅) ⋊ C₄²
derived, Sylow	H(3,𝔽₅) ⋊ C₂²×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#14522510
commutator, hypocenter	H(3,𝔽₅)	C₂²×C₄²
Frattini	C₅	C₂²×AGL(1,𝔽₅)²
Fitting	C₂²×H(3,𝔽₅)	C₄²
socle	C₂²×C₅	AGL(1,𝔽₅)²
prime radical	C₂²×((C₅⋊D₅)⋊C₅)⋊C₂	C₂²
cosocle	((C₅⋊D₅)⋊C₅)⋊C₂	C₂⁴
LCS	G ▷ H(3,𝔽₅)
UCS	C₁ ◁ C₂²

1797 conjugacy classes of subgroups; only the 313 normal subgroups are shown.

normal subgroups only

This group has 1797 conjugacy classes of subgroups — too many to draw. Only its 313 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 100

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

C₂²×AGL(2,𝔽₅)

GAP

Past the SmallGroups library coverage — no SmallGroups id.