D₄×#92302 #72228567

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

D₄×#92302

Splittings

Fitting	D₄×#6891 ⋊ C₂
Sylow	#6891 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂×C₃	C₂²×C₃⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₃²⋊C₉	C₂³×C₃
Frattini, socle	C₂×C₃²	C₂²×C₃×C₃⋊D₃
Fitting	D₄×#6891	C₂
hypercenter	C₃×D₄	C₃⋊AGL(1,ℤ/9)
hypocenter	C₃²⋊C₉	C₂×C₃×D₄
LCS	G ▷ C₂×C₃²⋊C₉ ▷ C₃²⋊C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₃×D₄

1792 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1792 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 22
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 22 points:

(1,2)(3,7,5,9,8,6)(11,13)(14,18)(15,17);(11,14,15)(13,17,18)(20,22);(1,5,9,4,3,6,2,8,7)(10,11,13)(12,15,17)(14,18,16)(19,20,21,22)

Conjugacy classes & character table

195 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#13091120, D₄×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃
maximal normal subgroup of	D₄×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃

GAP

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

gap> G := Group((1,2)(3,7,5,9,8,6)(11,13)(14,18)(15,17), (11,14,15)(13,17,18)(20,22), (1,5,9,4,3,6,2,8,7)(10,11,13)(12,15,17)(14,18,16)(19,20,21,22));