#71584

order	384
isoclinism	isoclinic class of #4656
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1818

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator, socle	C₂³×C₃	C₂²×C₄
Fitting	C₂×C₃×#117	C₂
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#117	D₃
hypocenter	C₃	#1818
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 192

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

(1,2,3)(5,7)(8,10,11,14)(9,13,12,15)(18,24)(21,27)(25,30)(28,31);(9,12)(13,15)(16,18)(17,21)(19,25)(20,24)(22,28)(23,27)(26,30)(29,31);(2,3)(4,5)(6,7)(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=f,c^2=d,f^2=g,b^a=bd,c^a=ce,h^a=h^2,c^b=cg

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682507, #10682511, #10687366, …
maximal normal subgroup of	#10682507, #10682511, #10687366, …
maximal quotient of	#10688522, #10688525, #10688528, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4160);