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D₃×#97 #4704

order	192
automorphism	Aut(G) = D₃×#394058 (order 3072) · Inn(G) = C₂²×D₃ · Out(G) = #2403
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₃×#97 ⋊ C₂
Sylow	C₃ ⋊ C₂×#97

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂×C₄²
Fitting	C₃×#97	C₂
socle	C₂²×C₃	C₂²×C₄
prime radical	C₂²×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#97	D₃
hypocenter	C₃	C₂×#97
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#329, #68341, #68345, …
maximal normal subgroup of	D₃×#329, #68341, #68345, …
maximal quotient of	D₃×#322, #68069, #68070, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 263);