#4800

order	192
automorphism	Aut(G) = C₂⁵×D₃×D₄ (order 1536) · Inn(G) = D₃×D₄ · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#103 ⋊ C₂
Sylow	C₃ ⋊ #488

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#103	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₁₂	C₂²
hypercenter	#103	D₃
hypocenter	C₃	#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 96

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

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

Polycyclic presentation:

2^6,3|a^2=e,b^2=e,c^2=f,d^2=f,b^a=bef,c^a=cef,g^a=g^2,c^b=cd,d^b=df,d^c=df

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#4800, #78785, #78807, …
maximal normal subgroup of	C₂×#4800, #78785, #78807, …
maximal quotient of	#68804, #68844, #68846, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 359);