C₄×C₃₁×C₂²⋊Q₈ #16842235

order	3968
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₄×C₃₁×C₂²⋊Q₈

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂³
commutator	C₂²	C₂³×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂⁴
prime radical	C₂⁴×C₃₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 4 generators on 47 points:

(1,2,3,4)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47);(6,8)(9,10,12,14)(11,15,16,13);(9,11,12,16)(10,13,14,15);(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)

Conjugacy classes & character table

1736 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#7566, C₃₁×#7583, C₃₁×#7589, …
maximal normal subgroup of	C₃₁×#7566, C₃₁×#7583, C₃₁×#7589, …
maximal quotient of	C₄×C₃₁×#385, C₄×C₃₁×#387, C₄×C₃₁×#389, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1034);

C4×C31×C22⋊Q8 #16842235