C₂×D₄×Q₈×C₃₁ #16843399

order	3968
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₁ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×D₄×Q₈×C₃₁

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 1984

A minimal faithful permutation representation — 5 generators on 45 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₁×#8379, C₃₁×#8388, C₃₁×#8391, …
maximal normal subgroup of	C₃₁×#8379, C₃₁×#8388, C₃₁×#8391, …
maximal quotient of	C₂×Q₈×C₂²⋊C₄×C₃₁, C₂×D₄×C₄⋊C₄×C₃₁, C₂×Q₈×C₄⋊C₄×C₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3968, 2198);

C2×D4×Q8×C31 #16843399