#71461

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2177

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting, prime radical	C₂×C₃×D₁₆	C₂
socle	C₂²×C₃	C₂×D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×D₁₆	D₃
hypocenter	C₃	#2177
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

172 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

2^7,3|a^2=e,c^2=eg,d^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fg,h^a=h^2,d^b=df,f^b=fg,h^c=h^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10715813, #10715815, D₃×#2177, …
maximal normal subgroup of	#10715813, #10715815, D₃×#2177, …
maximal quotient of	#10706938, #10707111, #10707195, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4037);