Finite Group Databasebeta

← Prev (#403) Next (#405) →

C₄×M₁₆ #404

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = #72219157 (order 1024) · Inn(G) = C₂² · Out(G) = #35680
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

C₄×M₁₆

Subgroups

	subgroup	G/H
center	C₄²	C₂²
commutator	C₂	C₂×C₄²
Frattini, cosocle	C₂×C₄	C₂³
socle	C₂²	C₂²×C₄
prime radical	C₂³	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

71 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 12 points:

(1,2,3,4);(5,6,7,10,8,9,11,12);(6,9)(10,12)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#143162, #143163, #143164, …
Frattini subgroup of	#7225, #7226, C₂×#7225, …
maximal subgroup of	#1278, #1283, #1288, …
maximal normal subgroup of	#1278, #1283, #1288, …
maximal quotient of	C₈×M₁₆, #1451, #1453, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 85);