#69046

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2254

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	#2254
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 384

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

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

Polycyclic presentation:

2^7,3|a^2=d,b^2=d,c^2=efg,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	#10682846, #10682848, #10709151, …
maximal normal subgroup of	#10682846, #10682848, #10709151, …
maximal quotient of	#10711624, #10713222, #10713788, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 1622);