C₂²⋊(Q₈⋊C₉) #63702

order	288
automorphism	?
homology	Abelianization (H₁) = C₉ · Schur multiplier (H₂) = C₂³

Aliases

C₂²⋊(Q₈⋊C₉)

C₂²×Q₈ ⋊ C₉

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₃	C₂²⋊A₄
commutator, hypocenter	C₂²×Q₈	C₉
Fitting, cosocle	C₂²×C₃×Q₈	C₃
socle, prime radical	C₂³×C₃	A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂×C₃

73 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

3^2,2^5|a^3=b,e^2=g,f^2=g,c^a=d,d^a=cd,e^a=f,f^a=ef,f^e=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 350);