#83776

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₃×M₁₆ ⋊ C₂
Sylow	C₃ ⋊ #3056

Subgroups

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5791	C₂
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#3056
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×M₁₆

496 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)(10,17)(13,22)(16,23)(18,21);(5,7);(8,10,11,18,12,17,20,21)(9,13,14,16,15,22,19,23);(2,3)(4,5)(6,7)(8,9,12,15)(10,16,17,23)(11,19,20,14)(13,21,22,18)

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10832621, #10832626, #10832643, …
maximal normal subgroup of	#10832621, #10832626, #10832643, …
maximal quotient of	#10777966, #10777988, #10778010, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16352);