C₂²⋊(C₄²⋊C₃) #5461

order	192
isoclinism	isoclinic class of C₂²⋊(C₄²⋊C₃)
automorphism	Aut(G) order 73728
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂³×C₄

Aliases

C₂²⋊(C₄²⋊C₃)

C₂²×C₄² ⋊ C₃

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂²×C₄²	C₃
Frattini	C₂²	C₂²⋊A₄
socle	C₂⁴	A₄
LCS	G ▷ C₂²×C₄²
UCS	C₁

103 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 3 generators on 16 points:

(2,3,4)(5,6,9)(7,11,16)(8,12,15)(10,14,13);(1,4)(2,3);(5,7,8,13)(6,12)(9,16,15,14)(10,11)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1020);