#83750

order	384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #2905

Subgroups

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃
commutator, socle	C₂²×C₃	C₂³×C₄
Fitting	C₂²×C₃×M₁₆	C₂
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂×C₃×C₄	C₂⁴
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#2905
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

484 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10701229, #10701231, #10701232, …
maximal normal subgroup of	#10701229, #10701231, #10701232, …
maximal quotient of	#10764331, #10764335, #10764339, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16326);