C₄×Q₈⋊C₉ #63424

order	288
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂²×C₃×S₄ (order 288) · Inn(G) = A₄ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₄×C₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄×Q₈⋊C₉
unambiguous semidirect product	(C₄×Q₈)⋊C₉

derived	Q₈ ⋊ C₄×C₉
Sylow	C₄×Q₈ ⋊ C₉

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	A₄
commutator, hypocenter	Q₈	C₄×C₉
Frattini, socle, prime radical	C₂²×C₃	C₂×A₄
Fitting	C₃×C₄×Q₈	C₃
cosocle	C₂×C₃×Q₈	C₂×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₄

49 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 72);