#83559

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×D₄×Q₈ ⋊ C₂
Sylow	C₃ ⋊ #3266

Subgroups

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×D₄×Q₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	#5550	C₂
hypercenter	D₄×Q₈	D₃
hypocenter	C₃	#3266
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ D₄×Q₈

494 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)(10,17)(11,21)(13,22)(14,20)(16,23)(18,19);(5,7);(8,10,12,19)(9,13,15,23)(11,17,21,18)(14,22,20,16);(2,3)(4,5)(6,7)(8,9,12,15)(10,16,19,22)(11,20,21,14)(13,17,23,18)

Polycyclic presentation:

2^7,3|a^2=g,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=h^2,d^b=de,e^b=eg,e^d=eg,h^d=h^2

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#3266, C₂×#83559, #11670263, …
maximal normal subgroup of	D₃×#3266, C₂×#83559, #11670263, …
maximal quotient of	#10784565, #10784685, #10784820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 16135);