#71829

order	384
isoclinism	isoclinic class of #4537
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1544

Subgroups

	subgroup	G/H
center	C₂×C₄	#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#533	D₃
hypocenter	C₃	#1544
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10686195, #10686197, #10686205, …
maximal normal subgroup of	#10686195, #10686197, #10686205, …
maximal quotient of	#10683142, #10683159, #10683488, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4405);