D₄×#2110 #72320006

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

D₄×#2110

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×C₄²
commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂³×D₄	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#33471, D₄×#33469
maximal normal subgroup of	D₄×#33471, D₄×#33469

GAP

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

gap> G := Group((1,2)(3,4), (5,6,8,13,10,14,22,29,11,15,9,23,21,30,24,36)(7,16,18,32,20,26,34,28,17,31,19,35,33,12,25,27), (5,7,11,17)(6,12)(8,19,9,18)(10,20,21,33)(13,28)(14,16)(15,26)(22,25,24,34)(23,27)(29,35)(30,31)(32,36), (7,17)(12,26)(16,31)(18,19)(20,33)(25,34)(27,28)(32,35), (2,4));

D4×#2110 #72320006