#67716

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1307

Subgroups

	subgroup	G/H
center	C₂²	C₃⋊#95
commutator, prime radical	C₂³×C₃	C₄²
Frattini	C₂×C₄²	C₂×D₃
Fitting	C₃×#423	C₂
socle	C₂²×C₃	#95
cosocle	C₂×C₃×C₄²	C₂²
hypercenter	#423	D₃
hypocenter	C₃	#1307
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

130 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682733, #10682734, #10682811, …
maximal normal subgroup of	#10682733, #10682734, #10682811, …
maximal quotient of	#10681761, #10681763, #10681779, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 292);