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C₃×#41⋊C₉ #88690

order	432
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₉ · Schur multiplier (H₂) = C₃

solvable

Aliases

direct product	C₃×#41⋊C₉
unambiguous semidirect product	(C₃×#41)⋊C₉

Splittings

#41 ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃²×C₄	A₄
commutator, hypocenter	Q₈	C₂×C₃×C₉
Frattini	C₂×C₃	C₂×C₃×A₄
Fitting, prime radical	C₃²×#41	C₃
socle	C₂×C₃²	C₂×A₄
cosocle	C₃×Q₈	C₂×C₃²
LCS	G ▷ Q₈
UCS	C₁ ◁ C₃²×C₄

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13146355, #13146360, #13146361, …
maximal subgroup of	C₃×#63422, C₃⋊#63422, C₃⋊(#131⋊C₉), …
maximal normal subgroup of	C₃×#63422, C₃⋊#63422, C₃⋊(#131⋊C₉), …
maximal quotient of	C₃×C₄×Q₈⋊C₉, C₂×C₃×#41⋊C₉, C₃×#41⋊C₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 337);