C₂×#72221828 #72244004

order	26244
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂×#72221828

Splittings

Fitting	C₂×#72221818 ⋊ C₂
Sylow	#72221818 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72221828
commutator, cosocle, hypocenter	#14550423	C₂²×C₃
Frattini	C₃²×C₉	C₂×C₃×C₃⋊(C₃⋊D₃)
Fitting	C₂×#72221818	C₂
socle	C₂×C₃	#72221827
prime radical	C₂×#72221819	C₃
LCS	G ▷ #14550423
UCS	C₁ ◁ C₂

3606 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72221873
maximal normal subgroup of	C₂×#72221873
holomorph of	C₂×C₂₇⋊C₃

GAP

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

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

C2×#72221828 #72244004