#71980

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1729

Subgroups

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂³×C₃	C₂²×C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1729
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

416 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(14,20)(17,23)(21,26)(24,27);(2,3)(4,5,6,7)(9,11)(12,14)(13,17)(15,21)(16,20)(18,24)(19,23)(22,26)(25,27);(2,3)(8,9)(10,11)(12,13,15,18,16,19,22,25)(14,17,21,24,20,23,26,27)

Polycyclic presentation:

2^7,3|a^2=e,b^2=f,e^2=g,b^a=bd,h^a=h^2,c^b=cg,h^b=h^2

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682155, #10682159, #10700538, …
maximal normal subgroup of	#10682155, #10682159, #10700538, …
maximal quotient of	#10684568, #10684573, #10684576, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4556);