Finite Group Databasebeta

← Prev (#2110) Next (#2112) →

C₂×C₂²⋊C₁₆ #2111

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×#72214599 (order 2048) · Inn(G) = C₂² · Out(G) = C₂²×#3025
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂×C₂²⋊C₁₆

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂²
commutator	C₂	C₂²×C₁₆
Frattini, cosocle	C₂×C₈	C₂³
socle	C₂³	C₂×C₈
prime radical	C₂⁴	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

136 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 22 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#93669, #93670, #93681, …
maximal subgroup of	#7183, #7186, #7199, …
maximal normal subgroup of	#7183, #7186, #7199, …
maximal quotient of	C₂×#1314, C₂×#1322, #11459, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 843);