#3025

order	128
nilpotency class	3
isoclinism	isoclinic class of B(4,𝔽₂)
automorphism	Aut(G) = C₂×#72218857 (order 3072) · Inn(G) = #120 · Out(G) = C₂²×S₄
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#120
commutator, Frattini, cosocle	C₂³	C₂⁴
socle	C₂	C₂×#120
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^7|d^2=g,b^a=be,c^a=cf,f^b=fg,e^c=eg

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#3025⋊C₃, #11711588, #11711590, …
maximal subgroup of	#13045, #13046, #21483, …
maximal normal subgroup of	#13045, #13046, #21483, …
maximal quotient of	#18122, #18123, #18124, …
automorphism group of	#349

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1757);