C₂²⋊(C₂²⋊A₄) #5982

order	192
isoclinism	isoclinic class of C₂²⋊(C₂²⋊A₄)
automorphism	Aut(G) order 23224320
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²⋊(C₂²⋊A₄)

C₂⁶ ⋊ C₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂⁶	C₃
LCS	G ▷ C₂⁶
UCS	C₁

1015 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 4 generators on 12 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1541);