Finite Group Databasebeta

← Prev (#1730) Next (#1732) →

C₂×#338 #1731

order	128
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄²
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	#95
prime radical	C₂⁴	Q₈
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2);(3,4,6,10,8,11,7,16)(5,12,14,9,15,17,13,18);(3,5,7,14,8,15,6,13)(4,9,10,12,11,18,16,17)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#338⋊C₃, #11710957, #11710958, …
maximal subgroup of	#7255, #7257, C₄×#338, …
maximal normal subgroup of	#7255, #7257, C₄×#338, …
maximal quotient of	C₂×#1275, #7860, #7861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 463);