#82336

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2353

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×C₄×D₄	C₂
socle, cosocle	C₂³×C₃	C₂⁴
prime radical	C₂²×C₃×D₄	C₂²
hypercenter	C₂×C₄×D₄	D₃
hypocenter	C₃	#2353
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10688740, #10688744, #10688745, …
maximal normal subgroup of	#10688740, #10688744, #10688745, …
maximal quotient of	#10716620, #10716644, #10716652, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14912);