Finite Group Databasebeta

← Prev (#5825) Next (#5827) →

Dic₃×#41 #5826

order	192
automorphism	Aut(G) order 18432
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

Splittings

C₃ ⋊ C₄×#41

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₃
commutator	C₂×C₃	C₂³×C₄
Frattini	C₂²	C₂³×D₃
Fitting, prime radical	C₂×C₃×#41	C₂
socle, cosocle	C₂²×C₃	C₂⁴
hypercenter	C₂×#41	D₃
hypocenter	C₃	C₄×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

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:

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#70039, #70131, Dic₃×#104, …
maximal normal subgroup of	#70039, #70131, Dic₃×#104, …
maximal quotient of	Dic₃×#117, #82227, #82229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1385);