C₃₁×#1703 #16841636

order	3968
nilpotency class	3
isoclinism	isoclinic class of #1685
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃₁×#1703

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃₁
prime radical	C₂×D₄×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

154 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(32,33)(34,36)(35,37)(38,39)(40,41,44,47)(42,48,50,55)(43,52,49,46)(45,53,54,51);(33,36)(37,38)(40,42,49,53,44,50,43,51)(41,45,52,55,47,54,46,48);(33,37)(35,39)(36,38)(42,50)(45,54)(48,55)(51,53)

Conjugacy classes & character table

713 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#1703, C₃₁×#19142, C₃₁×#19158, …
maximal normal subgroup of	C₂×C₃₁×#1703, C₃₁×#19142, C₃₁×#19158, …
maximal quotient of	C₃₁×#8656, C₃₁×#9223, C₃₁×#9227, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 435);