#82324

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2318

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	#2318
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

568 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)(9,13)(12,14)(16,18,19,22)(17,21,20,23);(10,15)(12,14)(17,20)(21,23);(8,10)(9,12)(11,15)(13,14);(2,3)(4,5,6,7)(8,9)(10,14)(11,13)(12,15)(16,17)(18,21)(19,20)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10762712, #10762716, #10762727, …
maximal normal subgroup of	#10762712, #10762716, #10762727, …
maximal quotient of	#10716456, #10716463, #10716472, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 14900);