C₃₁×#2175 #16842108

order	3968
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₁ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₁×#2175

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×D₈
commutator	C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂³×C₃₁
prime radical	C₂×D₈×C₃₁	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ C₄×C₈×C₃₁ ◁ G

160 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 1984

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)(32,33,35,34)(36,37)(38,41)(39,48)(40,42)(43,45)(44,46)(47,49)(50,51);(37,41)(38,45)(39,46)(42,49)(43,44)(47,51)(48,50);(32,34,35,33)(37,42)(41,49)(43,50)(44,48)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#12065, C₃₁×#12077, C₃₁×#12080, …
maximal normal subgroup of	C₃₁×#12065, C₃₁×#12077, C₃₁×#12080, …
maximal quotient of	C₃₁×#12043, C₃₁×#12074, C₃₁×#12075, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 907);