C₂×D₅₈₄₁ #37552498

order	23364
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₁₁₆₈₂
direct product	C₂×D₅₈₄₁
unambiguous semidirect product	(C₂×C₉)⋊D₆₄₉
unambiguous semidirect product	(C₂×C₁₁)⋊D₅₃₁
unambiguous semidirect product	(C₂×C₅₉)⋊D₉₉

Splittings

Fitting	C₂×C₉×C₁₁×C₅₉ ⋊ C₂
derived, Hall	C₉×C₁₁×C₅₉ ⋊ C₂²
Hall	C₁₁×C₅₉ ⋊ C₂×D₉
Hall	C₉×C₅₉ ⋊ C₂×D₁₁
Hall	C₉×C₁₁ ⋊ C₂×D₅₉
Sylow	C₅₉ ⋊ C₂×D₉₉
Sylow	C₁₁ ⋊ C₂×D₅₃₁
Sylow	C₉ ⋊ C₂×D₆₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₅₈₄₁
commutator, cosocle, hypocenter	C₉×C₁₁×C₅₉	C₂²
Frattini	C₃	C₂×D₁₉₄₇
Fitting	C₂×C₉×C₁₁×C₅₉	C₂
socle	C₂×C₃×C₁₁×C₅₉	D₃
LCS	G ▷ C₉×C₁₁×C₅₉
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 11682

A minimal faithful permutation representation — 2 generators on 81 points:

(1,2)(3,4,5,6,7,8,9,10,11)(12,13,14,15,16,17,18,19,20,21,22)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(4,11)(5,10)(6,9)(7,8)(13,22)(14,21)(15,20)(16,19)(17,18)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

5844 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₃₆₄
maximal normal subgroup of	D₂₃₃₆₄
maximal quotient of	Dic₁₁₆₈₂, D₂₃₃₆₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23364, 18);

C2×D5841 #37552498