D₄×#13091119 #72229345

order	11664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³×C₃

Aliases

direct product

D₄×#13091119

Splittings

Fitting	D₄×#10624749 ⋊ C₂
Sylow	#10624749 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂²×#92300
commutator, cosocle	C₂×C₃²⋊H(3,𝔽₃)	C₂³×C₃
Frattini	C₂×C₃³	C₂²×C₃×C₃⋊D₃
Fitting	D₄×#10624749	C₂
hypercenter	C₃²×D₄	C₃⋊((C₃⋊D₃)⋊C₃)
hypocenter	C₃²⋊H(3,𝔽₃)	C₂×C₃×D₄
LCS	G ▷ C₂×C₃²⋊H(3,𝔽₃) ▷ C₃²⋊H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃²×D₄

3092 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:

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

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#72210336
maximal normal subgroup of	D₄×#72210336

GAP

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

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

D4×#13091119 #72229345