Finite Group Databasebeta

← Prev (#63637) Next (#63639) →

C₂×C₃⋊(C₃⋊C₁₆) #63638

order	288
isoclinism	isoclinic class of C₃⋊D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product	C₂×C₃⋊(C₃⋊C₁₆)
unambiguous semidirect product	(C₂×C₃)⋊(C₃⋊C₁₆)

Splittings

C₃² ⋊ C₂×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₈	C₃⋊D₃
commutator, hypocenter	C₃²	C₂×C₁₆
Frattini	C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃²×C₈	C₂
socle, prime radical	C₂²×C₃²	C₈
cosocle	C₃²×C₈	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₈

84 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 3 generators on 24 points:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₃×C₃⋊C₁₆, #10589103, (C₂×D₃)⋊(C₃⋊C₁₆), …
maximal normal subgroup of	Dic₃×C₃⋊C₁₆, #10589103, (C₂×D₃)⋊(C₃⋊C₁₆), …
maximal quotient of	C₄×C₃⋊(C₃⋊C₁₆), C₄⋊(C₃⋊(C₃⋊C₁₆)), C₂×C₃⋊(C₃⋊C₃₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 286);