C₃×C₁₉×C₃₂ #14268779

order	1824
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³×C₈×C₉ (order 576) · Inn(G) = C₁ · Out(G) = C₂³×C₈×C₉
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₈₂₄
abelian	C₃×C₁₉×C₃₂

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂×C₃×C₁₉
socle, prime radical	C₂×C₃×C₁₉	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 1824

A minimal faithful permutation representation — 1 generator on 54 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35)(36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54)

Conjugacy classes & character table

1824 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₁₉×C₆₄, C₂×C₃×C₁₉×C₃₂, C₉×C₁₉×C₃₂, …
maximal normal subgroup of	C₃×C₁₉×C₆₄, C₂×C₃×C₁₉×C₃₂, C₉×C₁₉×C₃₂, …
maximal quotient of	C₃×C₁₉×C₆₄, C₂×C₃×C₁₉×C₃₂, C₉×C₁₉×C₃₂, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1824, 6);

C3×C19×C32 #14268779