C₃₁×#2825 #16842758

order	3968
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₃₁×#2825

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

522 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 168 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 — 4 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)(33,36)(37,39)(40,41)(42,45)(43,44)(46,47)(48,49,52,53)(50,54,55,51);(34,38)(37,39)(41,44)(42,46)(50,55)(51,54);(32,33)(34,37)(35,36)(38,39)(40,41)(42,47)(43,44)(45,46)(48,50)(49,51)(52,55)(53,54);(32,34)(33,37)(35,38)(36,39)(40,42)(41,45)(43,46)(44,47)(48,51)(49,50)(52,54)(53,55)

Conjugacy classes & character table

806 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₁×#2825, C₃₁×#36670, C₃₁×#36676, …
maximal normal subgroup of	C₂×C₃₁×#2825, C₃₁×#36670, C₃₁×#36676, …
maximal quotient of	C₃₁×#14621, C₃₁×#14637, C₃₁×#14642, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1557);