C₈²⋊C₃ #4444

order	192
isoclinism	isoclinic class of C₈²⋊C₃
automorphism	Aut(G) = #72245429 (order 6144) · Inn(G) = G · Out(G) = #123
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₈

Aliases

C₈²⋊C₃

C₈² ⋊ C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₈²	C₃
Frattini	C₄²	A₄
socle	C₂²	C₄²⋊C₃
LCS	G ▷ C₈²
UCS	C₁

19 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

3,2^6|b^2=de,c^2=dg,d^2=fg,e^2=f,b^a=c,c^a=bce,d^a=efg,e^a=de,f^a=g,g^a=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 3);