Finite Group Databasebeta

← Prev (#88905) Next (#88907) →

C₃×C₂²⋊(C₂²⋊C₉) #88906

order	432
isoclinism	isoclinic class of C₂²⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₉ · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

direct product	C₃×C₂²⋊(C₂²⋊C₉)
unambiguous semidirect product	(C₂²×C₃)⋊(C₂²⋊C₉)

Splittings

C₂⁴ ⋊ C₃×C₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	C₂²⋊A₄
commutator, hypocenter	C₂⁴	C₃×C₉
Frattini	C₃	C₃×C₂²⋊A₄
Fitting, socle, prime radical	C₂⁴×C₃²	C₃
cosocle	C₂⁴×C₃	C₃²
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₃²

190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of	C₃⋊#64188, C₃⋊#10588305, #13146405, …
maximal subgroup of	C₃⋊(#120⋊C₉), C₃×#120⋊C₉, C₃×#64188, …
maximal normal subgroup of	C₃⋊(#120⋊C₉), C₃×#120⋊C₉, C₃×#64188, …
maximal quotient of	C₃×C₂²⋊(Q₈⋊C₉), C₃×#63703, C₂×C₃×C₂²⋊(C₂²⋊C₉), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 553);