C₂×#14515305 #72267196

order	3840
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

direct product

C₂×#14515305

Subgroups

	subgroup	G/H
center, socle, hypercenter	C₂²	#11757163
commutator, hypocenter	#14515305	C₂
Frattini	C₂	C₂×#11757163
Fitting	C₂×2₋¹⁺⁴	A₅
cosocle	2₋¹⁺⁴	C₂×A₅
LCS	G ▷ #14515305
UCS	C₁ ◁ C₂²

283 conjugacy classes of subgroups; only the 9 normal subgroups are shown.

normal subgroups only

This group has 283 conjugacy classes of subgroups — too many to draw. Only its 9 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) = 82
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 82 points:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72267232, #72267528, #72268202, …

GAP

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

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