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Q₈×C₂²⋊C₉ #63698

order	288
automorphism	Aut(G) = C₃×S₄² (order 1728) · Inn(G) = C₂²×A₄ · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₂²×C₉ · Schur multiplier (H₂) = C₂

Aliases

Q₈×C₂²⋊C₉

Splittings

C₂²×Q₈ ⋊ C₉

Subgroups

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

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13164025, #13164031, SL(2,𝔽₃)×(C₂²⋊C₉)⋊C₂, …
maximal subgroup of	#10589518, #10589519, SD₁₆×C₂²⋊C₉, …
maximal normal subgroup of	#10589518, #10589519, SD₁₆×C₂²⋊C₉, …
maximal quotient of	C₄⋊C₄×C₂²⋊C₉, #557⋊C₉, Q₈×Q₈⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 346);