#79856

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #3389

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₃×D₄
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#442	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂×C₃×D₈	C₂²
hypercenter	#442	D₃
hypocenter	C₃	#3389
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

348 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)(4,7,8,5)(6,9,11,10)(14,22)(17,21)(23,27)(25,26);(13,18)(14,22)(15,24)(17,25)(19,20)(21,26);(4,6,8,11)(5,9,7,10)(12,14,16,22)(13,17,20,21)(15,23,24,27)(18,26,19,25);(2,3)(4,5,8,7)(6,10,11,9)(12,13)(14,21)(15,18)(16,20)(17,22)(19,24)(23,25)(26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

51 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3389, C₂×#79856, #11675742, …
maximal normal subgroup of	D₃×#3389, C₂×#79856, #11675742, …
maximal quotient of	#10811859, #10811863, #10813481, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 12432);