#71421

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2151

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle	C₂²×C₃	C₂²×C₈
Fitting	C₂×C₃×M₃₂	C₂
prime radical	C₂³×C₃	C₂×C₈
cosocle	C₂×C₃×C₈	C₂³
hypercenter	C₂×M₃₂	D₃
hypocenter	C₃	#2151
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(10,18)(14,24)(19,30)(20,31)(25,34)(26,35)(32,38)(36,39);(2,3)(4,5,6,7)(8,10)(9,14)(11,19)(12,20)(13,18)(15,25)(16,26)(17,24)(21,32)(22,30)(23,31)(27,36)(28,34)(29,35)(33,38)(37,39);(2,3)(5,7)(8,9,11,15,12,16,21,27,13,17,22,28,23,29,33,37)(10,14,19,25,20,26,32,36,18,24,30,34,31,35,38,39)

Polycyclic presentation:

2^7,3|a^2=e,b^2=d,e^2=f,f^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	#10682591, #10711323, #10711326, …
maximal normal subgroup of	#10682591, #10711323, #10711326, …
maximal quotient of	#10706736, #10706762, #10706796, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 3997);