C₂²×Q₈⋊C₉ #63697

order	288
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₃×S₄² (order 1728) · Inn(G) = A₄ · Out(G) = C₂×C₃×S₄
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

C₂²×Q₈⋊C₉

derived	Q₈ ⋊ C₂²×C₉
Sylow	C₂²×Q₈ ⋊ C₉

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

101 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 345);