Finite Group Databasebeta

← Prev (#72244271) Next (#72244273) →

C₃×D₇×B(4,𝔽₂) #72244272

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

C₃×D₇×B(4,𝔽₂)

Splittings

Fitting	C₃×C₇×B(4,𝔽₂) ⋊ C₂
Hall	C₃×C₇ ⋊ C₂×B(4,𝔽₂)
Sylow	C₇ ⋊ C₂×C₃×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂×C₃	D₇×#120
commutator, cosocle	C₂³×C₇	C₂⁴×C₃
Frattini	C₂³	C₂³×C₃×D₇
Fitting	C₃×C₇×B(4,𝔽₂)	C₂
socle	C₂×C₃×C₇	C₂×#120
hypercenter	C₃×B(4,𝔽₂)	D₇
hypocenter	C₇	C₂×C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ C₃×B(4,𝔽₂)

2036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 168

A minimal faithful permutation representation — 4 generators on 18 points:

(1,2,4,6,7,5,3)(8,9,10)(11,13)(12,15)(14,17)(16,18);(2,3)(4,5)(6,7);(15,18);(11,12,14,16)(13,15,17,18)

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×D₇×B(4,𝔽₂), #72246128, C₃×AGL(1,𝔽₇)×B(4,𝔽₂), …
maximal normal subgroup of	D₃×D₇×B(4,𝔽₂), C₃×AGL(1,𝔽₇)×B(4,𝔽₂)
maximal quotient of	#72246128, B(4,𝔽₂)×ΓL(1,𝔽₈)⋊D₇

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,4,6,7,5,3)(8,9,10)(11,13)(12,15)(14,17)(16,18), (2,3)(4,5)(6,7), (15,18), (11,12,14,16)(13,15,17,18));