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C₄²×SL(2,𝔽₃) #73164

order	384
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₄

solvable

Aliases

C₄²×SL(2,𝔽₃)

Splittings

Fitting, Sylow	C₄²×Q₈ ⋊ C₃
derived	Q₈ ⋊ C₃×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄²	A₄
commutator, hypocenter	Q₈	C₃×C₄²
Frattini, socle	C₂³	C₂²×A₄
Fitting	C₄²×Q₈	C₃
prime radical	C₂²×SL(2,𝔽₃)	C₂²
cosocle	C₂²×Q₈	C₂²×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₄²

229 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 16 points:

(5,6,7,8)(10,11,13)(14,16,15);(1,2,3,4)(9,13,12,15)(10,16,14,11)

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×C₈×SL(2,𝔽₃), SL(2,𝔽₃)×#97, #20569⋊C₃, …
maximal normal subgroup of	C₄×C₈×SL(2,𝔽₃), SL(2,𝔽₃)×#97, #20569⋊C₃, …
maximal quotient of	SL(2,𝔽₃)×#95, C₄×C₈×SL(2,𝔽₃), SL(2,𝔽₃)×#97, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5740);