Finite Group Databasebeta

← Prev (#5846) Next (#5848) →

C₃×C₄×#41 #5847

order	192
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 6144
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

C₃×C₄×#41

Subgroups

	subgroup	G/H
center	C₃×C₄²	C₂²
commutator	C₂	C₂³×C₃×C₄
Frattini, cosocle	C₂²	C₂⁴×C₃
socle	C₂²×C₃	C₂⁴
prime radical	C₂×C₃×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ G

310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#71638, #71639, #71640, …
maximal subgroup of	#71638, #71639, #71640, …
maximal normal subgroup of	#71638, #71639, #71640, …
maximal quotient of	C₃×C₄×#117, C₃×C₄²×D₄, C₃×C₄²×Q₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1406);