C₃₁×#3415 #16843348

order	3968
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₁×#3415

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ #41×C₃₁ ◁ C₃₁×#131 ◁ G

370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 63
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 4 generators on 63 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)(34,39)(35,38)(36,49)(37,42)(40,46)(41,47)(43,53)(44,50)(45,59)(48,52)(51,60)(54,58)(55,57)(56,61)(62,63);(33,38)(35,46)(36,47)(39,50)(40,41)(42,52)(44,58)(45,57)(48,60)(49,51)(53,61)(54,55)(56,63)(59,62);(32,34)(33,39)(35,44)(36,45)(37,43)(38,50)(40,54)(41,55)(42,53)(46,58)(47,57)(48,56)(49,59)(51,62)(52,61)(60,63);(34,43)(39,53)(44,56)(45,57)(50,61)(54,62)(55,59)(58,63)

Conjugacy classes & character table

992 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₁×#13669, C₃₁×#13671, C₃₁×#33854, …
maximal normal subgroup of	C₃₁×#13669, C₃₁×#13671, C₃₁×#33854, …
maximal quotient of	C₃₁×#33381, C₃₁×#33410, C₃₁×#33454, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2147);