#87192

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×AGL(1,ℤ/8) ⋊ C₂
Sylow	C₃ ⋊ #3585

Subgroups

	subgroup	G/H
center	C₂	C₂²×D₃×D₄
commutator, cosocle	C₃×C₄	C₂⁵
Frattini	C₄	C₂⁴×D₃
Fitting	C₂×C₃×AGL(1,ℤ/8)	C₂
socle	C₂×C₃	C₂³×D₄
hypercenter	C₂×AGL(1,ℤ/8)	D₃
hypocenter	C₃	#3585
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

1464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 19 points:

(1,2,3)(5,11)(9,12)(10,17)(16,18);(6,13)(10,17)(14,19)(16,18);(2,3)(4,6)(5,10)(7,14)(8,13)(9,16)(11,17)(12,18)(15,19);(5,9)(7,15)(10,16)(11,12)(14,19)(17,18);(4,5)(6,10)(7,9)(8,11)(12,15)(13,17)(14,16)(18,19)

Polycyclic presentation:

2^7,3|f^2=g,b^a=bf,e^a=eg,f^a=fg,f^b=fg,d^c=dg,h^c=h^2

Conjugacy classes & character table

63 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10826474, #10826476, #10826496, …
maximal normal subgroup of	#10826474, #10826476, #10826496, …
maximal quotient of	#11658755, #11658767, #11658868, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 19768);