#72227488

order	34560
isoclinism	isoclinic class of #72227488
automorphism	?
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂×C₃	#72209614
commutator, hypocenter	G	C₁
Fitting, cosocle	C₂⁵×C₃	A₆
LCS	G
UCS	C₁ ◁ C₂×C₃

385 conjugacy classes of subgroups; only the 7 normal subgroups are shown.

normal subgroups only

This group has 385 conjugacy classes of subgroups — too many to draw. Only its 7 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) = 98
transitive degree	μ_t(G) = 432

A minimal faithful permutation representation — 2 generators on 98 points:

(1,43,58,76,9)(2,57,54,15,10)(3,7,28,26,34)(4,70,78,56,22)(5,59,35,65,71)(6,24,38,20,41)(8,74,45,52,27)(11,64,42,69,17)(12,19,79,47,49)(13,29,55,62,72)(14,63,25,77,61)(16,31,68,53,44)(18,39,48,51,21)(23,73,46,40,66)(30,37,50,75,32)(33,36,60,80,67)(81,97,88,93,84)(82,94,90,96,85)(83,95,92,98,86);(1,54,14,68,72)(2,6,29,61,9)(3,19,69,16,46)(4,26,30,80,71)(5,76,52,44,39)(7,17,59,32,22)(8,41,36,21,10)(11,63,27,48,34)(12,23,15,24,37)(13,56,49,33,66)(18,65,75,31,58)(20,67,70,45,51)(25,74,64,55,28)(35,57,50,79,38)(40,62,77,60,78)(42,53,43,47,73)(81,98,88,91,84,83,96,92,89,86,82,93,90,87,85)(94,95,97)

Conjugacy classes & character table

58 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((1,43,58,76,9)(2,57,54,15,10)(3,7,28,26,34)(4,70,78,56,22)(5,59,35,65,71)(6,24,38,20,41)(8,74,45,52,27)(11,64,42,69,17)(12,19,79,47,49)(13,29,55,62,72)(14,63,25,77,61)(16,31,68,53,44)(18,39,48,51,21)(23,73,46,40,66)(30,37,50,75,32)(33,36,60,80,67)(81,97,88,93,84)(82,94,90,96,85)(83,95,92,98,86), (1,54,14,68,72)(2,6,29,61,9)(3,19,69,16,46)(4,26,30,80,71)(5,76,52,44,39)(7,17,59,32,22)(8,41,36,21,10)(11,63,27,48,34)(12,23,15,24,37)(13,56,49,33,66)(18,65,75,31,58)(20,67,70,45,51)(25,74,64,55,28)(35,57,50,79,38)(40,62,77,60,78)(42,53,43,47,73)(81,98,88,91,84,83,96,92,89,86,82,93,90,87,85)(94,95,97));