#401

order	64
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	Aut(G) order 10752
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

59 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^6|a^2=def,b^2=de,c^2=d,b^a=bd,c^a=ce,c^b=cf

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#401⋊C₇, C₂×#401⋊C₇, C₃×#401⋊C₇, …
Fitting subgroup of	#401⋊C₃, #401⋊C₇, (#401⋊C₇)⋊C₃
maximal subgroup of	C₂×#401, #2401, #2493, …
maximal normal subgroup of	C₂×#401, #2401, #2493, …
maximal quotient of	#1443, #2104, C₂×#401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(64, 82);