#63700

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

solvable

Aliases

No aliases for this group.

Splittings

Sylow

H(5,𝔽₂) ⋊ C₉

Subgroups

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂²×A₄
commutator, hypocenter	Q₈	C₂²×C₉
Fitting, prime radical	C₃×H(5,𝔽₂)	C₃
cosocle	C₃×Q₈	C₂²×C₃
hypercenter	C₃×Q₈	A₄
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

81 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 33
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 33 points:

(1,11,21,9,4,19,6,18,12,3,15,24)(2,16,20,5,8,23,10,14,17,7,13,22)(25,29,33,27,32,28,30,26,31);(3,9)(4,14)(5,7)(8,18)(11,13)(12,24)(15,16)(17,23)(19,21)(20,22)

Polycyclic presentation:

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

Conjugacy classes & character table

57 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13164041, #13164044, #13164045, …
maximal subgroup of	#10589530, #10589531, #576⋊C₉, …
maximal normal subgroup of	#10589530, #10589531, #576⋊C₉, …
maximal quotient of	#520⋊C₉, C₄⋊(#41⋊C₉), Q₈×Q₈⋊C₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 348);