C₃₁×#1773 #16841706

order	3968
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₁ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₁×#1773

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×D₄
prime radical	C₂³×C₃₁	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₈×C₃₁ ◁ G

224 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 51
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 3 generators on 51 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)(37,41,43,46)(38,44)(42,49,50,48)(45,51);(32,33,34,35)(36,37)(38,42)(39,46)(40,43)(41,47)(44,50)(45,49)(48,51);(36,38,39,45,40,44,47,51)(37,42,46,49,43,50,41,48)

Conjugacy classes & character table

1364 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#12647, C₃₁×#12648, C₃₁×#12649, …
maximal normal subgroup of	C₃₁×#12647, C₃₁×#12648, C₃₁×#12649, …
maximal quotient of	C₃₁×#8268, C₃₁×#8269, C₃₁×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 505);