C₂×#14546335 #72243130

order	4374
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₂×#14546335

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#10624462
commutator	C₂₇	C₂×C₃×C₂₇
Frattini, cosocle	#6864	C₂×C₃²
prime radical	C₂×C₃²	#6875
LCS	G ▷ C₂₇ ▷ C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₉ ◁ C₂×#6864 ◁ G

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 83 points:

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

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#72221799
maximal subgroup of	C₂×#72221799
maximal normal subgroup of	C₂×#72221799

GAP

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

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

C2×#14546335 #72243130