#71496

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2176

Subgroups

	subgroup	G/H
center	C₂²	D₃×D₈
commutator	C₃×C₈	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₂×C₃×Q₃₂	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×Q₃₂	D₃
hypocenter	C₃	#2176
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2)(4,6,7,5)(9,15)(14,29)(16,31)(17,27)(18,35)(22,34)(30,33)(32,38);(8,10,13,21)(9,14,15,29)(11,23,26,39)(12,28,24,20)(16,32,31,38)(17,33,27,30)(18,34,35,22)(19,36,37,25);(2,3)(4,5,7,6)(8,9)(10,18)(11,22)(12,27)(13,15)(14,19)(16,25)(17,24)(20,38)(21,35)(23,33)(26,34)(28,32)(29,37)(30,39)(31,36)

Polycyclic presentation:

2^7,3|a^2=e,b^2=g,c^2=e,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	#10715806, #10715808, D₃×#2176, …
maximal normal subgroup of	#10715806, #10715808, D₃×#2176, …
maximal quotient of	#10706933, #10707132, #10707190, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4072);