C₂×C₄×Q₈×C₃₁ #14520697

order	1984
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₁ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₄×Q₈×C₃₁

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂²
commutator	C₂	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²	C₂⁴×C₃₁
socle, prime radical	C₂³×C₃₁	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 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);(7,8,9,10)(11,14,13,12);(7,11,9,13)(8,12,10,14)

Conjugacy classes & character table

1240 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	#16844035, #16844055, C₂×Q₈⋊(C₃₁⋊C₈), …
maximal subgroup of	C₃₁×#1461, C₃₁×#1467, C₂×C₃₁×Q₈⋊C₈, …
maximal normal subgroup of	C₃₁×#1461, C₃₁×#1467, C₂×C₃₁×Q₈⋊C₈, …
maximal quotient of	C₂×C₄×C₄⋊C₄×C₃₁, C₄²×Q₈×C₃₁, C₂×C₃₁×#387, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1984, 197);

C2×C4×Q8×C31 #14520697