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C₃×#827 #63613

order	288
isoclinism	isoclinic class of #826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₁

Aliases

Splittings

C₃² ⋊ SD₃₂

Subgroups

	subgroup	G/H
center	C₂×C₃	#213
commutator, cosocle	C₃×C₈	C₂²×C₃
Frattini	C₈	C₂×C₃×D₃
Fitting, prime radical	C₃²×D₈	C₂
socle	C₂×C₃²	D₈
hypercenter	C₃×D₈	D₃
hypocenter	C₃	C₃×SD₃₂
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×C₈ ◁ C₃×D₈

63 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 2 generators on 22 points:

(1,2,3)(4,5,6)(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20);(5,6)(7,15,11,8)(9,21,18,12)(10,14,17,19)(13,16,20,22)

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#827, #10591028, #10591029, …
maximal normal subgroup of	D₃×#827, #10591028, #10591029, …
maximal quotient of	C₃×#4490, C₃×#4492, C₃×#4562, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 261);