C₂×#72221921 #72269394

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

C₂×#72221921

Splittings

Fitting	C₂×#14555440 ⋊ C₂
derived	#6908 ⋊ C₂²×C₃²
Sylow	#14555440 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221921
commutator, cosocle, hypocenter	#6908	C₂²×C₃²
Frattini	C₃²	C₂×C₃²×C₃⋊(C₃⋊D₃)
Fitting	C₂×#14555440	C₂
socle	C₂×C₃	C₃²×C₃⋊#272
LCS	G ▷ #6908
UCS	C₁ ◁ C₂

8484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 162

A minimal faithful permutation representation — 4 generators on 83 points:

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

Conjugacy classes & character table

380 conjugacy classes. The character table is too large to display.

GAP

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

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

C2×#72221921 #72269394