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C₂×#357 #2136

order	128
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	D₈
commutator	C₈	C₂²×C₄
Frattini, cosocle	C₂×C₈	C₂³
prime radical	C₂²×D₈	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ 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)(10,19)(11,15)(13,16)(14,17)(18,21)(20,22);(8,12)(9,16)(10,17)(13,14)(15,21)(18,22)(19,20)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#357, #12637, #12731, …
maximal normal subgroup of	C₄×#357, #12637, #12731, …
maximal quotient of	C₂×#1331, #11645, #11657, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 868);