#4697

order	192
automorphism	Aut(G) = D₃×#392109 (order 3072) · Inn(G) = C₂×D₁₂ · Out(G) = #379
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₄×C₈ ⋊ C₂
Sylow	C₃ ⋊ #488

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂³
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×C₄×C₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×D₁₂	C₂²
hypercenter	C₄×C₈	D₃
hypocenter	C₃	#488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

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

Polycyclic presentation:

2^6,3|b^2=ef,c^2=de,d^2=f,b^a=be,c^a=cd,d^a=df,g^a=g^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#4697, #78463, #78469, …
maximal normal subgroup of	C₂×#4697, #78463, #78469, …
maximal quotient of	#68630, #68637, #68638, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 256);