C₁₉⋊C₁₆ #64512

order	304
isoclinism	isoclinic class of D₁₉
automorphism	?
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₁₉⋊C₁₆

Splittings

derived, Sylow

C₁₉ ⋊ C₁₆

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₁₉
commutator, hypocenter	C₁₉	C₁₆
Fitting, cosocle	C₈×C₁₉	C₂
socle, prime radical	C₂×C₁₉	C₈
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₈

10 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 2 generators on 35 points:

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

Polycyclic presentation:

2^4,19|a^2=b,b^2=c,c^2=d,e^a=e^18

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×D₁₉, #10597255, C₂×C₁₉⋊C₁₆, …
maximal normal subgroup of	C₁₆×D₁₉, #10597255, C₂×C₁₉⋊C₁₆, …
maximal quotient of	C₁₉⋊C₃₂, C₂×C₁₉⋊C₁₆, C₃×C₁₉⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 1);

C19⋊C16 #64512