#72209775

order	21504
isoclinism	isoclinic class of #72209775
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁷ ⋊ GL(3,𝔽₂)

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#72209643
commutator, hypocenter	G	C₁
Fitting, cosocle	C₂⁷	GL(3,𝔽₂)
socle	C₂⁴	AGL(3,𝔽₂)
LCS	G
UCS	C₁ ◁ C₂

2608 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 2608 conjugacy classes of subgroups — too many to draw. Only its 8 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:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227627, #72227637

GAP

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

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