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C₄×Dic₃⋊C₄ #4931

order	192
isoclinism	isoclinic class of Dic₆
automorphism	Aut(G) order 6144
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

C₄×Dic₃⋊C₄

Splittings

C₃ ⋊ C₄×C₄⋊C₄

Subgroups

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

194 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 192

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

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#68611, #68614, #68617, …
maximal normal subgroup of	#68611, #68614, #68617, …
maximal quotient of	#69403, Dic₃⋊#95, #69406, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 490);