D₄×#6910 #14517825

order	1944
nilpotency class	3
isoclinism	isoclinic class of D₄×#6909
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product	D₄×#6910
unambiguous semidirect product	C₄⋊(C₂×#6910)

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₃×D₄×H(3,𝔽₃)	C₃
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃³×D₄ ◁ G

544 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 198 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₄×#4184⋊C₃, D₄×#92283, D₄×#4177⋊C₃, …
maximal subgroup of	D₄×#10624749, D₄×#4184⋊C₃, D₄×#92283, …
maximal normal subgroup of	D₄×#10624749, D₄×#4184⋊C₃, D₄×#92283, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2219);