#71987

order	384
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #1742

Subgroups

	subgroup	G/H
center	C₂×C₄	(C₂×D₃)⋊C₄
commutator	C₂²×C₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂²×C₃×M₁₆	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂²×M₁₆	D₃
hypocenter	C₃	#1742
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

388 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10700399, #10700400, #10700538, …
maximal normal subgroup of	#10700399, #10700400, #10700538, …
maximal quotient of	#10684777, #10684780, #10684789, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 4563);