Finite Group Databasebeta

← Prev (#67496) Next (#67498) →

C₁₆×C₃⋊C₈ #67497

order	384
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₈×C₁₆ · Schur multiplier (H₂) = C₈

Aliases

C₁₆×C₃⋊C₈

Splittings

C₃ ⋊ C₈×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₁₆	D₃
commutator, hypocenter	C₃	C₈×C₁₆
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₄×C₈
cosocle	C₃×C₄×C₈	C₂²
LCS	G ▷ C₃
UCS	C₁ ◁ C₄×C₁₆

104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃⋊C₈×C₃₂, #10682514, #10682519, …
maximal normal subgroup of	C₃⋊C₈×C₃₂, #10682514, #10682519, …
maximal quotient of	C₁₆×C₃⋊C₁₆, #10681802, #10682014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 73);