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C₂×#99 #409

order	64
nilpotency class	3
isoclinism	isoclinic class of #99
automorphism	Aut(G) = #6499874 (order 512) · Inn(G) = C₂²⋊C₄ · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊C₄
commutator	C₂²	C₂²×C₄
Frattini, cosocle	C₂³	C₂³
prime radical	C₂²×D₄	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

105 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 3 generators on 10 points:

(1,2);(3,4)(5,9,10,7)(6,8);(3,5)(4,7)(6,10)(8,9)

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#1390, #1393, C₄×#99, …
maximal normal subgroup of	#1390, #1393, C₄×#99, …
maximal quotient of	C₂×#323, C₂×#324, #1459, …
outer automorphism group of	#540, #541

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 90);