#72209659

order	10752
isoclinism	isoclinic class of #72209659
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	G	C₁
Frattini, Fitting, socle, cosocle	C₂⁶	GL(3,𝔽₂)
LCS	G
UCS	C₁

884 conjugacy classes of subgroups; only the 5 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 5 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)(15,16)(17,20)(18,21)(22,26)(24,27)(25,28);(2,4,5)(6,7,9)(8,10,11)(12,13,14)(16,18,19)(20,22,23)(21,24,25)(26,27,28);(2,6)(5,9)(10,13)(11,14);(2,6)(4,7)(8,12)(11,14);(1,3)(8,12)(10,13)(11,14);(16,20)(19,23)(21,26)(24,27);(18,22)(19,23)(21,26)(25,28);(15,17)(21,26)(24,27)(25,28)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72209871

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