D₄×#3484 #72233503

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

D₄×#3484

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

145397 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

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

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233731, #72234468, #72234898, …
maximal normal subgroup of	#72233731, #72234468, #72234898, …
maximal quotient of	#72233580, D₄²×#120, #72240174, …

GAP

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

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

D4×#3484 #72233503