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C₂×#97 #403

order	64
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = #72219167 (order 2048) · Inn(G) = C₂² · Out(G) = #449475
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

Subgroups

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

73 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 14 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#93229, #93230, #93231, …
Frattini subgroup of	#7105, #7109, #7110, …
maximal subgroup of	#1275, #1281, #1295, …
maximal normal subgroup of	#1275, #1281, #1295, …
maximal quotient of	C₂×#322, #1451, #1452, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 84);