C₂×#72221916 #72244383

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

Aliases

direct product

C₂×#72221916

Splittings

Fitting	C₂×#14555437 ⋊ C₂
derived	C₃²×C₉ ⋊ C₂²×C₃³
Sylow	#14555437 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221916
commutator, cosocle, hypocenter	C₃²×C₉	C₂²×C₃³
Frattini	C₃²	C₂×C₃³×C₃⋊D₃
Fitting	C₂×#14555437	C₂
socle	C₂×C₃	C₃²×C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃²×C₉
UCS	C₁ ◁ C₂

7644 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of

C₂×#72221920

GAP

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

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

C2×#72221916 #72244383