D₄×#10624749 #72228533

order	5832
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃³ · Schur multiplier (H₂) = C₂×C₃⁵

Aliases

direct product

D₄×#10624749

Subgroups

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

1952 conjugacy classes of subgroups.

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₄×#13091120, D₄×#13091119, D₄×(#4169⋊C₃)⋊C₃, …
maximal subgroup of	D₄×#13091120, D₄×#13091119, S₄×#10624749, …
maximal normal subgroup of	D₄×#13091120, D₄×#13091119, D₄×(#4169⋊C₃)⋊C₃

GAP

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

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

D4×#10624749 #72228533