D₄×#92285 #72229338

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

D₄×#92285

Splittings

Fitting	D₄×#6911 ⋊ C₂
Sylow	#6911 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₂×H(3,𝔽₃)	C₂³×C₃²
Frattini	C₂×C₃²	C₂²×C₃²×D₃
Fitting	D₄×#6911	C₂
hypercenter	C₃²×D₄	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₂×C₃²×D₄
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×D₄

964 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 964 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#13091119, D₄×(#4178⋊C₂)⋊C₃
maximal normal subgroup of	D₄×(#4178⋊C₂)⋊C₃

GAP

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

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