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AGL(1,𝔽₇)×#87830 #72243139

order	16800
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

AGL(1,𝔽₇)×#87830

Splittings

derived, Hall	C₅²×C₇ ⋊ C₂×C₃×C₄²
Hall	D₇×#87830 ⋊ C₃
Hall	C₅²×ΓL(1,𝔽₈) ⋊ C₂×C₄²
Sylow	C₅² ⋊ C₄²×AGL(1,𝔽₇)
Hall	ΓL(1,𝔽₈) ⋊ C₂×#87830
Sylow	C₇ ⋊ C₂×C₃×#87830

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	AGL(1,𝔽₇)×D₅⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅²×C₇	C₂×C₃×C₄²
Fitting, socle	C₂×C₅²×C₇	C₂²×C₃×C₄
prime radical	C₂×AGL(1,𝔽₇)×C₅⋊D₅	C₂²
cosocle	C₂×C₇×C₅⋊D₅	C₂³×C₃
LCS	G ▷ C₅²×C₇
UCS	C₁ ◁ C₂

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 140

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#11717375
maximal normal subgroup of	AGL(1,𝔽₇)×#11717375

GAP

Past the SmallGroups library coverage — no SmallGroups id.