#72238840

order	61440
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72209997
commutator, hypocenter	G	C₁
Frattini	C₂⁶	#11757162
Fitting, cosocle	C₂²×C₄⁴	A₅
socle	C₂⁵	#14515300
LCS	G
UCS	C₁ ◁ C₂

9848 conjugacy classes of subgroups; only the 10 normal subgroups are shown.

normal subgroups only

This group has 9848 conjugacy classes of subgroups — too many to draw. Only its 10 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 84
transitive degree	μ_t(G) = 120

A minimal faithful permutation representation — 2 generators on 84 points:

(1,5,9,10)(2,11)(3,4,19,12)(6,17,16,24)(7,20)(8,14)(13,18)(15,23,22,21)(25,47,41,58,30,32,29,48)(26,45,33,39)(27,83)(28,68,40,46,31,75,38,69)(34,65,35,44)(36,64)(37,63,57,73,43,80,59,77)(42,78)(49,71,52,72)(50,84,62,67,53,81,56,82)(51,66,54,70)(60,79);(1,2,5)(3,7,4)(6,14,23)(8,10,9)(11,21,16)(12,19,18)(13,22,17)(15,24,20)(25,26,31)(27,34,49)(28,30,33)(29,39,38)(32,44,63)(35,52,42)(36,54,71)(37,55,62)(40,41,45)(43,61,56)(46,67,79)(47,65,80)(48,70,73)(50,59,76)(51,72,60)(53,57,74)(58,66,77)(64,69,82)(68,81,78)(75,84,83)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,5,9,10)(2,11)(3,4,19,12)(6,17,16,24)(7,20)(8,14)(13,18)(15,23,22,21)(25,47,41,58,30,32,29,48)(26,45,33,39)(27,83)(28,68,40,46,31,75,38,69)(34,65,35,44)(36,64)(37,63,57,73,43,80,59,77)(42,78)(49,71,52,72)(50,84,62,67,53,81,56,82)(51,66,54,70)(60,79), (1,2,5)(3,7,4)(6,14,23)(8,10,9)(11,21,16)(12,19,18)(13,22,17)(15,24,20)(25,26,31)(27,34,49)(28,30,33)(29,39,38)(32,44,63)(35,52,42)(36,54,71)(37,55,62)(40,41,45)(43,61,56)(46,67,79)(47,65,80)(48,70,73)(50,59,76)(51,72,60)(53,57,74)(58,66,77)(64,69,82)(68,81,78)(75,84,83));