D₄×#58990 #72335255

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

D₄×#58990

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 11458 conjugacy classes of subgroups; only the 40936 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 16 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72335839

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((6,8)(14,16), (5,6)(7,8)(13,14)(15,16), (6,8)(10,11), (2,4)(10,11), (2,4), (9,10,12,11)(13,14)(15,16), (1,2)(3,4)(5,6)(7,8)(9,11,12,10));

D4×#58990 #72335255