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C₂×#4427 #67371

order	378
isoclinism	isoclinic class of #4427
automorphism	Aut(G) = C₃×D₃×AGL(1,𝔽₇) (order 756) · Inn(G) = C₃×ΓL(1,𝔽₈) · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₁

Aliases

Splittings

Fitting	C₂×C₇×C₉ ⋊ C₃
Hall	C₂×C₇ ⋊ 3₋¹⁺²
Sylow	C₇ ⋊ C₂×3₋¹⁺²

Subgroups

	subgroup	G/H
center	C₂×C₃	C₃×ΓL(1,𝔽₈)
commutator, cosocle	C₃×C₇	C₂×C₃²
Frattini	C₃	C₂×C₃×ΓL(1,𝔽₈)
Fitting	C₂×C₇×C₉	C₃
socle	C₂×C₃×C₇	C₃²
prime radical	C₂×C₃×ΓL(1,𝔽₈)	C₃
hypercenter	C₂×C₉	ΓL(1,𝔽₈)
hypocenter	C₇	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₉

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 18 points:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10625387, #10625391, #10625397, …
maximal normal subgroup of	#10625387, #10625391, #10625397, …
maximal quotient of	C₄×#4427, C₂²×#4427, C₂×(C₇⋊C₉)⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(378, 24);