#41⋊SL(2,𝔽₃) #85653

order	384
automorphism	Aut(G) = C₂²×A₄⋊S₄ (order 1152) · Inn(G) = C₂²⋊A₄ · Out(G) = C₂²×D₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

solvable

Aliases

unambiguous semidirect product

#41⋊SL(2,𝔽₃)

Splittings

Fitting, Sylow

Q₈×#41 ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	C₂²⋊A₄
commutator, cosocle, hypocenter	Q₈²	C₂×C₃
Frattini, socle	C₂²	C₂×C₂²⋊A₄
Fitting	Q₈×#41	C₃
LCS	G ▷ Q₈²
UCS	C₁ ◁ C₂×C₄

223 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(2,3,5)(6,8,7)(9,10,11,15)(12,18,20,22,14,17,19,24,13,16,21,23);(1,2,4,6)(3,7,8,5)(9,13)(10,17)(11,20)(12,21)(14,19)(15,23)(16,24)(18,22);(1,5,4,7)(2,8,6,3)(13,20)(14,21)(17,23)(18,24)

Polycyclic presentation:

2,3,2^6|a^2=g,c^2=gh,d^2=g,e^2=h,f^2=g,c^b=defg,d^b=f,e^b=cefg,f^b=dfg,d^c=dg,e^c=eh,f^d=fg

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72268245, #72268253, #72268254, …
maximal subgroup of	#131⋊SL(2,𝔽₃), C₂×#41⋊SL(2,𝔽₃), #11710697, …
maximal normal subgroup of	#131⋊SL(2,𝔽₃), C₂×#41⋊SL(2,𝔽₃), #11710697, …
maximal quotient of	C₄×Q₈⋊SL(2,𝔽₃), #52287⋊C₃, #52291⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18229);