#78824

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #3084

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#416	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×#41	C₂²
hypercenter	#416	D₃
hypocenter	C₃	#3084
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

376 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 4 generators on 27 points:

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

Polycyclic presentation:

2^7,3|a^2=e,b^2=eg,d^2=g,f^2=g,b^a=beg,c^a=cf,f^a=fg,d^b=dg,h^b=h^2,f^c=fg

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10852727, #10852730, #10852736, …
maximal normal subgroup of	#10852727, #10852730, #10852736, …
maximal quotient of	#10794712, #10796189, #10796231, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11400);