#72209793

order	21504
isoclinism	isoclinic class of #72209793
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209646
commutator, hypocenter	G	C₁
Frattini, socle	C₂⁴	AGL(3,𝔽₂)
Fitting, cosocle	C₂⁷	GL(3,𝔽₂)
LCS	G
UCS	C₁ ◁ C₂

2191 conjugacy classes of subgroups; only the 8 normal subgroups are shown.

normal subgroups only

This group has 2191 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:

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

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72227745

GAP

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

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