Finite Group Databasebeta

← Prev (#60) Next (#62) →

C₃×C₈ #61

order	24
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂³ (order 8) · Inn(G) = C₁ · Out(G) = C₂³
holomorph	G ⋊ Aut(G) = D₃×AGL(1,ℤ/8)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₄
abelian	C₃×C₈

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂×C₃
socle, prime radical	C₂×C₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 1 generator on 11 points:

(1,2,3,4,5,6,7,8)(9,10,11)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

center of	C₃×M₃₂, C₃×#131, C₃×C₃⋊C₁₆, …
derived subgroup of	D₄₈, #800, Dic₂₄, …
Frattini subgroup of	C₉⋊C₁₆, C₉×C₁₆, C₁₆×D₉, …
Fitting subgroup of	C₃⋊C₁₆, D₃×C₈, ΓL(1,𝔽₂₅), …
maximal subgroup of	C₃⋊C₁₆, C₃×C₁₆, D₃×C₈, …
maximal normal subgroup of	C₃⋊C₁₆, C₃×C₁₆, D₃×C₈, …
maximal quotient of	C₃×C₁₆, C₂×C₃×C₈, C₈×C₉, …
outer automorphism group of	C₇₃⋊C₃, #88036

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24, 2);