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C₃×M₁₆ #222

order	48
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×D₄ (order 32) · Inn(G) = C₂² · Out(G) = C₂³
holomorph	G ⋊ Aut(G) = D₃×#23872
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

C₃×M₁₆

Subgroups

	subgroup	G/H
center	C₃×C₄	C₂²
commutator	C₂	C₂×C₃×C₄
Frattini, cosocle	C₄	C₂²×C₃
socle	C₂×C₃	C₂×C₄
prime radical	C₂²×C₃	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 11 points:

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

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#822, #823, #824, …
maximal subgroup of	#822, #823, #824, …
maximal normal subgroup of	#822, #823, #824, …
maximal quotient of	C₃×#97, C₃×C₂²⋊C₈, C₃×C₄⋊C₈, …
outer automorphism group of	C₅²⋊C₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 24);