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C₄×#131 #2874

order	128
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂²
commutator	C₂	C₂²×C₄²
Frattini, cosocle	C₂×C₄	C₂⁴
socle	C₂²	C₂³×C₄
prime radical	C₂×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72286336, #72287445, #72287450, …
Fitting subgroup of	C₄×#131⋊C₃, #11710990, #11710991, …
maximal subgroup of	#8184, #8185, #8186, …
maximal normal subgroup of	#8184, #8185, #8186, …
maximal quotient of	C₈×#131, #17320, #17328, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1606);