#78548

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₃×#405 ⋊ C₂
Sylow	C₃ ⋊ #2873

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator	C₂×C₃	C₂²×C₄²
Fitting	C₃×#405	C₂
socle	C₂²×C₃	C₂³×C₄
prime radical	C₂³×D₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	#2873
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #405

564 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 4 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10711421, #10792692, #10792889, …
maximal normal subgroup of	#10711421, #10792692, #10792889, …
maximal quotient of	#10760768, #10760769, #10760792, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11124);