#69045

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2253

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#2253
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

116 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3)(12,14,23,39,17,26,27,33,15,24,35,43,28,40,16,25)(13,19,29,38,21,34,32,36,18,31,42,41,30,37,20,22);(4,6,7,11)(5,8,9,10)(13,18)(19,31)(20,32)(21,30)(22,36)(29,42)(34,37)(38,41);(2,3)(4,5,7,9)(6,10,11,8)(12,13,15,18)(14,22,24,36)(16,29,27,42)(17,30,28,21)(19,33,31,25)(20,35,32,23)(26,41,40,38)(34,39,37,43)

Polycyclic presentation:

2^7,3|a^2=d,b^2=dg,c^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=ce,e^a=ef,f^a=fg,h^a=h^2

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682843, #10682845, #10682849, …
maximal normal subgroup of	#10682843, #10682845, #10682849, …
maximal quotient of	#10711628, #10713217, #10713222, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1621);