C₃₁×#2242 #16842175

order	3968
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₁ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃₁×#2242

Subgroups

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

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

normal subgroups only

This group has 138 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) = 71
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 3 generators on 71 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)(34,37)(35,36)(38,39)(40,41)(42,54)(43,48)(44,62)(45,65)(46,66)(47,53)(49,58)(50,55)(51,64)(52,63)(56,68)(57,67)(59,71)(60,70)(61,69);(33,36)(37,39)(40,42,55,60,46,57,64,61,47,58,45,56,63,59,44,43)(41,48,62,71,52,68,65,49,53,69,51,67,66,70,50,54);(32,34,35,38)(33,37,36,39)(40,43,44,59,63,56,45,58,47,61,64,57,46,60,55,42)(41,49,50,68,66,71,51,48,53,54,65,70,52,67,62,69)

Conjugacy classes & character table

1178 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₁×#2242, C₃₁×#33697, C₃₁×#33702, …
maximal normal subgroup of	C₂×C₃₁×#2242, C₃₁×#33697, C₃₁×#33702, …
maximal quotient of	C₃₁×#12889, C₃₁×#13147, C₃₁×#13148, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 974);