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C₃×C₄×C₂²⋊C₄ #5251

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂×C₃×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

C₃×C₄×C₂²⋊C₄

Subgroups

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²
commutator	C₂	C₂×C₃×C₄²
Frattini, cosocle	C₂³	C₂³×C₃
socle	C₂³×C₃	C₂³
prime radical	C₂⁴×C₃	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

258 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)(13,14,15);(5,6,7,8)(9,10)(11,12);(10,12)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#67616, #67617, #67618, …
maximal subgroup of	#67616, #67617, #67618, …
maximal normal subgroup of	#67616, #67617, #67618, …
maximal quotient of	C₃×C₄×#95, C₃×C₂²⋊#95, C₃×#1434, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 810);