#71777

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1558

Subgroups

	subgroup	G/H
center	C₂²	C₂×#217
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	#1558
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

228 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)(13,18)(17,26)(19,23)(20,27);(5,8)(6,10)(12,14,16,22)(13,17,18,26)(15,24,25,21)(19,27,23,20);(2,3)(4,5,6,9,7,8,10,11)(12,13)(14,20)(15,23)(16,18)(17,21)(19,25)(22,27)(24,26)

Polycyclic presentation:

2^7,3|a^2=e,b^2=g,d^2=g,e^2=fg,b^a=bd,c^a=cg,d^a=df,h^a=h^2,d^b=dg,e^b=efg

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10682919, #10682920, #10687579, …
maximal normal subgroup of	#10682919, #10682920, #10687579, …
maximal quotient of	#10683276, #10683315, #10683422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4353);