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D₃×#127 #5714

order	192
automorphism	Aut(G) order 36864
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₃×#127 ⋊ C₂
Sylow	C₃ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#127	C₂
hypercenter	#127	D₃
hypocenter	C₃	C₂×#127
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #127

498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 4 generators on 11 points:

(1,2,3)(8,9,10,11);(4,5,6,7);(5,7)(9,11);(2,3)

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#331, #68383, #68387, …
maximal normal subgroup of	D₃×#331, #68383, #68387, …
maximal quotient of	#74319, #74328, #74334, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1273);