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D₃×#90016 #72244282

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

Splittings

Fitting	C₃×C₇×H(5,𝔽₂) ⋊ C₂²
Hall	C₃×C₇ ⋊ C₂×B(4,𝔽₂)
Sylow	C₇ ⋊ D₃×B(4,𝔽₂)
Sylow	C₃ ⋊ C₂×#90016

Subgroups

	subgroup	G/H
center	C₂	D₃×#6516
commutator, cosocle	C₂³×C₃×C₇	C₂⁴
Frattini	C₂³	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×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

2036 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has 2036 conjugacy classes of subgroups — too many to draw. Only its 208 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) = 168

Conjugacy classes & character table

201 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₇×B(4,𝔽₂), D₃×#13052144
maximal normal subgroup of	D₃×D₇×B(4,𝔽₂), D₃×#13052144

GAP

Past the SmallGroups library coverage — no SmallGroups id.