#78479

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

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #3162

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×#97	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×#97	D₃
hypocenter	C₃	#3162
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10690140, #10696177, #10696178, …
maximal normal subgroup of	#10690140, #10696177, #10696178, …
maximal quotient of	#10795610, #10798540, #10799764, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11055);