#78549

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 ⋊ C₄×#131

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₃
commutator	C₂×C₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#405	C₂
socle	C₂²×C₃	C₂³×C₄
prime radical	C₂×#235	C₂²
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	#405	D₃
hypocenter	C₃	C₄×#131
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #405

572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 192

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10710234, #10710239, #10711123, …
maximal normal subgroup of	#10710234, #10710239, #10711123, …
maximal quotient of	#10760774, C₈×#907, #10760798, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 11125);