C₂×#72221892 #72244384

order	8748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

direct product

C₂×#72221892

Splittings

Fitting	C₂×#14555436 ⋊ C₂
derived	C₃⁴ ⋊ C₂²×C₃³
Sylow	#14555436 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221892
commutator, cosocle, hypocenter	C₃⁴	C₂²×C₃³
Frattini	C₃²	C₂×C₃³×C₃⋊D₃
Fitting	C₂×#14555436	C₂
socle	C₂×C₃	C₃²×C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₂

7906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 83 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of

C₂×#72221898

GAP

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

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

C2×#72221892 #72244384