#857⋊C₂ #5397

order	192
isoclinism	isoclinic class of #857⋊C₂
automorphism	Aut(G) = #4626⋊C₂ (order 384) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

Fitting	#127 ⋊ D₃
derived	C₄²⋊C₃ ⋊ C₂²

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₄²⋊C₃	C₂²
Frattini, socle	C₂²	C₂×S₄
Fitting	#127	D₃
LCS	G ▷ C₄²⋊C₃
UCS	C₁

83 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 12

A minimal faithful permutation representation — 2 generators on 12 points:

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

Polycyclic presentation:

2^2,3,2^4|d^2=fg,e^2=f,c^a=c^2,d^a=e,e^a=d,f^a=fg,d^b=dfg,e^b=ef,d^c=e,e^c=deg,f^c=g,g^c=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 956);