C₃₁×#3428 #16843361

order	3968
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₃₁×#3428

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂⁴
commutator	C₂	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂²	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂⁵
prime radical	C₂²×#41×C₃₁	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ G

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 5 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)(32,33,34,35)(36,37)(38,44)(39,43)(40,42)(41,45)(46,50)(47,49)(48,51);(36,38)(37,41)(39,46)(40,45)(42,44)(43,48)(47,51)(49,50);(38,45)(41,44)(46,51)(48,50);(37,42)(41,44)(43,49)(48,50);(36,39,40,47)(37,43,42,49)(38,46,45,51)(41,48,44,50)

Conjugacy classes & character table

2108 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#10743, C₃₁×#10747, C₃₁×#10752, …
maximal normal subgroup of	C₃₁×#10743, C₃₁×#10747, C₃₁×#10752, …
maximal quotient of	C₃₁×#33996, C₃₁×#33999, C₄×C₃₁×#518, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2160);

C31×#3428 #16843361