Finite Group Databasebeta

← Prev (#1757) Next (#1759) →

C₄×#104 #1758

order	128
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = #72235118 (order 2048) · Inn(G) = D₄ · Out(G) = C₂×#2021
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₄²	D₄
commutator	C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄	C₂³
socle	C₂²	C₂×C₂²⋊C₄
prime radical	C₂×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

180 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);(5,6,8,11,9,7,12,10);(5,7)(6,9)(8,10)(11,12)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×#104, #8191, #8192, …
maximal normal subgroup of	C₈×#104, #8191, #8192, …
maximal quotient of	C₄×D₄⋊C₈, C₄×Q₈⋊C₈, #8172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 490);