#4799

order	192
automorphism	Aut(G) = C₂⁵×D₃×D₄ (order 1536) · Inn(G) = D₃×D₄ · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#103	C₂
socle, prime radical	C₂²×C₃	C₂×D₄
hypercenter	#103	D₃
hypocenter	C₃	#479
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(4,6,7,11)(5,8,9,10)(14,20)(16,23)(17,24)(19,27)(21,22)(25,26);(12,14,15,21)(13,17,18,25)(16,20,23,22)(19,24,27,26);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,18)(14,17,21,25)(16,19,23,27)(20,24,22,26)

Polycyclic presentation:

2^6,3|a^2=d,b^2=f,c^2=df,e^2=f,c^a=cdf,g^a=g^2,c^b=ce,e^b=ef,e^c=ef

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#4799, #78779, #78807, …
maximal normal subgroup of	C₂×#4799, #78779, #78807, …
maximal quotient of	#68797, #68802, #68807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 358);