#72209859

order	21504
isoclinism	isoclinic class of #72209859
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209657
commutator, hypocenter	G	C₁
Frattini	C₂⁴	AGL(3,𝔽₂)
Fitting, socle, cosocle	C₂⁷	GL(3,𝔽₂)
LCS	G
UCS	C₁ ◁ C₂

1924 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 1924 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Permutation generators:

(2,4)(3,7)(9,10)(11,14)(12,15)(16,20)(18,21)(19,22)(23,24,27,25)(26,29,28,30)(31,35,33,36)(32,37,34,38);(2,3,5)(6,8,7)(10,12,13)(14,16,17)(15,18,19)(20,21,22)(23,26,24)(25,27,28)(29,31,32)(30,33,34);(23,27)(24,25)(26,28)(29,30)(31,33)(32,34)(35,36)(37,38);(1,2)(3,5)(4,6)(7,8);(1,3)(2,5)(4,8)(6,7);(1,4)(2,6)(3,8)(5,7);(10,14)(13,17)(15,20)(18,21);(12,16)(13,17)(15,20)(19,22);(9,11)(15,20)(18,21)(19,22)

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227825

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

gap> G := Group((2,4)(3,7)(9,10)(11,14)(12,15)(16,20)(18,21)(19,22)(23,24,27,25)(26,29,28,30)(31,35,33,36)(32,37,34,38), (2,3,5)(6,8,7)(10,12,13)(14,16,17)(15,18,19)(20,21,22)(23,26,24)(25,27,28)(29,31,32)(30,33,34), (23,27)(24,25)(26,28)(29,30)(31,33)(32,34)(35,36)(37,38), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,8)(6,7), (1,4)(2,6)(3,8)(5,7), (10,14)(13,17)(15,20)(18,21), (12,16)(13,17)(15,20)(19,22), (9,11)(15,20)(18,21)(19,22));