C₂×D₁₀₈₁ #16864478

order	4324
automorphism	Aut(G) order 2187944
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₂₁₆₂
direct product	C₂×D₁₀₈₁
unambiguous semidirect product	(C₂×C₂₃)⋊D₄₇
unambiguous semidirect product	(C₂×C₄₇)⋊D₂₃

Splittings

Fitting, socle	C₂×C₂₃×C₄₇ ⋊ C₂
derived, Hall	C₂₃×C₄₇ ⋊ C₂²
Sylow	C₄₇ ⋊ C₂×D₂₃
Sylow	C₂₃ ⋊ C₂×D₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₁₀₈₁
commutator, cosocle, hypocenter	C₂₃×C₄₇	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₂₃×C₄₇	C₂
LCS	G ▷ C₂₃×C₄₇
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 2162

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25)(26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72);(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Conjugacy classes & character table

1084 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×D₁₀₈₁, C₂×D₃₂₄₃, C₂×C₃×D₁₀₈₁, …
maximal normal subgroup of	C₂×C₃×D₁₀₈₁, C₂×C₅×D₁₀₈₁, C₂×C₇×D₁₀₈₁, …
maximal quotient of	D₃×D₁₀₈₁, C₂×D₃₂₄₃, C₂×C₃×D₁₀₈₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4324, 11);

C2×D1081 #16864478