C₃₁×#3395 #16843328

order	3968
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₁ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₁×#3395

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ G

348 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 3968

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)(33,37)(38,39)(40,41,44,47)(42,48,50,55)(43,51,53,46)(45,52,54,49);(32,33,36,37)(34,38,35,39);(32,34,36,35)(33,38,37,39)(40,42,44,50)(41,45,47,54)(43,52,53,49)(46,55,51,48);(32,35,36,34)(33,38,37,39)

Conjugacy classes & character table

1085 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#12330, C₃₁×#12334, C₃₁×#12382, …
maximal normal subgroup of	C₃₁×#12330, C₃₁×#12334, C₃₁×#12382, …
maximal quotient of	C₃₁×#22542, Q₈×C₃₁×#107, C₃₁×#22808, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2127);