C₂₇⋊C₁₆ #88354

order	432
isoclinism	isoclinic class of D₂₇
automorphism	?
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₂₇⋊C₁₆

Splittings

derived, Sylow

C₂₇ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₈	D₂₇
commutator, hypocenter	C₂₇	C₁₆
Frattini	C₈×C₉	D₃
Fitting, cosocle	C₈×C₂₇	C₂
socle, prime radical	C₂×C₃	C₉⋊C₈
LCS	G ▷ C₂₇
UCS	C₁ ◁ C₈

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 432

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

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

Polycyclic presentation:

2^4,3^3|a^2=b,b^2=c,c^2=d,e^3=f^2g,f^3=g^2,e^a=e^2f,f^a=f^2g,g^a=g^2

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×D₂₇, #11720039, C₂×C₂₇⋊C₁₆, …
maximal normal subgroup of	C₁₆×D₂₇, #11720039, C₂×C₂₇⋊C₁₆, …
maximal quotient of	C₂₇⋊C₃₂, C₂×C₂₇⋊C₁₆, C₈₁⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 1);

C27⋊C16 #88354