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C₂×#322 #1448

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂²
commutator	C₂	C₂×C₄×C₈
Frattini, cosocle	C₄²	C₂³
socle, prime radical	C₂³	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

124 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#92819, #92820, #92839, …
maximal subgroup of	#7137, #7139, #7145, …
maximal normal subgroup of	#7137, #7139, #7145, …
maximal quotient of	C₄×#322, C₂×#1273, #8131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 180);