#2736

order	128
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	Aut(G) order 24576
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁴
prime radical	C₂⁵	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

280 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,16,12)(17,18,20,22)(19,23,24,21);(2,5)(3,8)(10,14)(12,15)(17,18,20,22)(19,21,24,23);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,21)(20,22)(23,24);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,21)(20,24)(22,23)

Polycyclic presentation:

2^7|a^2=eg,b^2=ef,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#2736⋊C₃, C₂×#2736⋊C₃, #52294⋊C₃, …
Fitting subgroup of	#2736⋊C₃, #11714254, #11714262
maximal subgroup of	#12188, #12264, #36725, …
maximal normal subgroup of	#12188, #12264, #36725, …
maximal quotient of	#14366, #14481, #14577, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(128, 1468);