Finite Group Databasebeta

← Prev (#1724) Next (#1726) →

C₄×#97 #1725

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²
commutator	C₂	C₄³
Frattini, cosocle	C₂²×C₄	C₂³
socle, prime radical	C₂³	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 16 points:

(1,2,3,4);(9,10,11,14,12,13,15,16);(5,6,7,8)(10,13)(14,16)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#92833, #92834, #92855, …
maximal subgroup of	#7019, #7030, #7149, …
maximal normal subgroup of	#7019, #7030, #7149, …
maximal quotient of	C₄×#322, C₈×#97, #7842, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 457);