Q₈×C₄⋊C₄×C₃₁ #16842283

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

Aliases

direct product

Q₈×C₄⋊C₄×C₃₁

Subgroups

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

568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 3968

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

(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8);(10,12)(13,14,15,16);(9,10,11,12)

Conjugacy classes & character table

1550 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×#8884, C₃₁×#8888, C₃₁×#8892, …
maximal normal subgroup of	C₃₁×#8884, C₃₁×#8888, C₃₁×#8892, …
maximal quotient of	C₄⋊C₄²×C₃₁, Q₈×C₃₁×#95, C₃₁×#13818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 1082);

Q8×C4⋊C4×C31 #16842283