C₂²×C₃₁×H(5,𝔽₂) #16843524

order	3968
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₃₁ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂²×C₃₁×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₃₁
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

6560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 43
transitive degree	μ_t(G) = 992

A minimal faithful permutation representation — 6 generators on 43 points:

(1,2)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(3,4);(5,6)(7,11)(8,9)(10,12);(6,9)(10,11);(5,6)(7,10)(8,9)(11,12);(5,7)(6,10)(8,12)(9,11)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#20585, C₃₁×#20586, C₃₁×#20592, …
maximal normal subgroup of	C₃₁×#20585, C₃₁×#20586, C₃₁×#20592, …
maximal quotient of	C₂²×C₃₁×#518, C₂²×C₃₁×#520, C₂×C₄×C₃₁×H(5,𝔽₂), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2323);

C22×C31×H(5,𝔽2) #16843524