#72209597

order	2688
isoclinism	isoclinic class of #72209597
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#13051525
commutator, hypocenter	G	C₁
Frattini, Fitting, socle, cosocle	C₂⁴	GL(3,𝔽₂)
LCS	G
UCS	C₁ ◁ C₂

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72227550, C₂×#72209597
maximal subgroup of	#72209785, #72209793, #72209804, …
maximal normal subgroup of	#72227550, C₂×#72209597
maximal quotient of	#72209785, #72209828, #72209859, …

GAP

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

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