#69806

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1746

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₂×C₃×#97	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1746
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 31 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10685643, #10685648, #10686232, …
maximal normal subgroup of	#10685643, #10685648, #10686232, …
maximal quotient of	#10685612, #10685617, #10685620, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 2382);