#85472

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

Aliases

No aliases for this group.

Splittings

Fitting

#577 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×S₄
commutator, cosocle	#41⋊C₃	C₂³
Frattini	C₄	C₂²×S₄
Fitting	#577	D₃
hypercenter	SD₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₂×D₄
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

248 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,42,23,17,24,43,7,15,26,3,5,41)(2,37,35,13,25,58,8,31,36,9,11,57)(4,39,50,14,45,61,18,47,52,19,20,62)(6,16,27,44)(10,55,59,30,51,64,28,53,60,29,33,63)(12,32,22,38)(21,48,46,40)(34,54,49,56);(2,8)(3,14)(4,18)(9,29)(11,25)(12,22)(13,30)(15,39)(16,40)(17,19)(20,45)(21,46)(31,53)(32,54)(35,36)(37,55)(38,56)(41,61)(42,47)(43,62)(44,48)(50,52)(57,63)(58,64);(1,2)(3,13)(4,10)(5,22)(6,25)(7,8)(9,17)(11,27)(12,24)(14,29)(15,32)(16,31)(18,28)(19,30)(20,49)(21,51)(23,36)(26,35)(33,46)(34,45)(37,44)(38,42)(39,56)(40,55)(41,57)(43,58)(47,54)(48,53)(50,60)(52,59)(61,64)(62,63)

Polycyclic presentation:

2^4,3,2^3|c^2=h,d^2=h,f^2=h,g^2=h,b^a=bh,c^a=ch,e^a=e^2,f^a=gh,g^a=fh,c^b=cd,d^b=dh,d^c=dh,f^e=g,g^e=fgh,g^f=gh

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#85472, #11714988, #11714989, …
maximal normal subgroup of	C₂×#85472, #11714988, #11714989, …
maximal quotient of	#11712365, #11712370, #11712373, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18048);