#69066

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2247

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	#2247
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

176 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 384

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682822, #10682824, #10682834, …
maximal normal subgroup of	#10682822, #10682824, #10682834, …
maximal quotient of	#10709845, #10709877, #10709902, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1642);