#85666

order	384
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#3573 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₈	C₂²⋊A₄
commutator, hypocenter	H(5,𝔽₂)	C₃×C₄
Frattini	C₄	C₂×C₂²⋊A₄
Fitting	#3573	C₃
socle	C₂	C₄×C₂²⋊A₄
prime radical	#5950	C₂
cosocle	#585	C₂×C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₈

279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2,3,2^6|a^2=c,c^2=h,d^2=h,f^2=h,d^b=dfgh,e^b=dgh,f^b=def,g^b=efh,g^d=gh,f^e=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72341787, #72341802, #72341830, …
maximal subgroup of	#12684⋊C₃, #21483⋊C₃, #21484⋊C₃, …
maximal normal subgroup of	#12684⋊C₃, #21483⋊C₃, #21484⋊C₃, …
maximal quotient of	#60191⋊C₃, #131⋊SL(2,𝔽₃), C₈×#997, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18242);