#85471

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

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

normal subgroups only

This group has 244 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:

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

Polycyclic presentation:

2^4,3,2^3|b^2=h,d^2=h,f^2=h,g^2=h,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

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#85471, #11714981, #11714993, …
maximal normal subgroup of	C₂×#85471, #11714981, #11714993, …
maximal quotient of	#11712271, #11712273, #11712278, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18047);