Finite Group Databasebeta

← Prev (#1309) Next (#1311) →

C₈×C₁₆ #1310

order	128
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = #72218016 (order 2048) · Inn(G) = C₁ · Out(G) = #72218016
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₈

Aliases

C₈×C₁₆

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₈	C₂²
socle, prime radical	C₂²	C₄×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);(17,18,19,20,21,22,23,24)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

center of	#93664, #93665, C₁₆×M₃₂, …
derived subgroup of	#72258200
Frattini subgroup of	C₁₆×C₃₂, #93664, #93665, …
maximal subgroup of	C₁₆², #7052, #7068, …
maximal normal subgroup of	C₁₆², #7052, #7068, …
maximal quotient of	C₁₆², #7052, #7136, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 42);