#83955

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₃×AGL(1,ℤ/8) ⋊ C₂
Sylow	C₃ ⋊ #3268

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×AGL(1,ℤ/8)	C₂
socle	C₂²×C₃	C₂²×D₄
hypercenter	C₂×AGL(1,ℤ/8)	D₃
hypocenter	C₃	#3268
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

(1,2,3)(9,15)(11,19)(13,16)(14,21)(17,23)(20,22);(2,3)(5,7)(8,10,12,18)(9,14,16,22)(11,17,19,23)(13,20,15,21);(8,9,12,16)(10,14,18,22)(11,15,19,13)(17,21,23,20);(4,5)(6,7)(9,13)(11,19)(14,20)(15,16)(17,23)(21,22)

Polycyclic presentation:

2^7,3|b^2=g,c^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,d^b=de,e^b=eg,h^c=h^2,e^d=eg

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3268, C₂×#83955, #11670258, …
maximal normal subgroup of	D₃×#3268, C₂×#83955, #11670258, …
maximal quotient of	#10784588, #10784708, #10784858, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16531);