#41⋊C₂₇ #88395

order	432
isoclinism	isoclinic class of SL(2,𝔽₃)
automorphism	Aut(G) = C₂×C₉×S₄ (order 432) · Inn(G) = A₄ · Out(G) = C₂²×C₉
homology	Abelianization (H₁) = C₂×C₂₇ · Schur multiplier (H₂) = C₁

solvable

Aliases

unambiguous semidirect product

#41⋊C₂₇

Splittings

Sylow

#41 ⋊ C₂₇

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₉	A₄
commutator, hypocenter	Q₈	C₂×C₂₇
Frattini	C₂×C₉	C₂×A₄
Fitting	C₉×#41	C₃
socle	C₂×C₃	C₂×C₂²⋊C₉
prime radical	C₃×#41	C₉
cosocle	Q₈×C₉	C₂×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₄×C₉

35 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,5,8)(3,7,11,14,4,13,10,16,9,6,12,15)(17,22,34,25,37,36,39,18,27,20,30,40,33,42,21,24,23,35,26,38,43,19,29,28,32,31,41);(1,9,5,11)(2,13,8,15)(3,12,10,4)(6,16,14,7)

Polycyclic presentation:

2,3^3,2^3|a^2=g,b^3=c,c^3=d,e^2=g,f^2=g,e^b=f,f^b=ef,f^e=fg

Conjugacy classes & character table

126 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#13136141, #13136144, #13136145, …
maximal subgroup of	#11720109, #131⋊C₂₇, #11720650, …
maximal normal subgroup of	#11720109, #131⋊C₂₇, #11720650, …
maximal quotient of	C₄×Q₈⋊C₂₇, C₂×#41⋊C₂₇, #41⋊C₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 42);

#41⋊C27 #88395