D₄×#35346 #72336077

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

D₄×#35346

Subgroups

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

858000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 20 points:

(10,12)(14,17)(15,19);(9,10)(11,12);(14,17)(15,19);(2,4)(6,8)(14,17)(15,19);(2,4)(6,8)(14,17)(18,20);(5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,6)(7,8)(13,15,16,19)(14,18,17,20)

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72336510

GAP

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

gap> G := Group((10,12)(14,17)(15,19), (9,10)(11,12), (14,17)(15,19), (2,4)(6,8)(14,17)(15,19), (2,4)(6,8)(14,17)(18,20), (5,6)(7,8)(9,10)(11,12)(13,14)(15,18)(16,17)(19,20), (1,2)(3,4)(5,6)(7,8)(13,15,16,19)(14,18,17,20));

D4×#35346 #72336077