C₂×D₄₅₄₅ #32911409

order	18180
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₉₀₉₀
direct product	C₂×D₄₅₄₅
unambiguous semidirect product	(C₂×C₅)⋊D₉₀₉
unambiguous semidirect product	(C₂×C₉)⋊D₅₀₅
unambiguous semidirect product	(C₂×C₁₀₁)⋊D₄₅

Splittings

Fitting	C₂×C₅×C₉×C₁₀₁ ⋊ C₂
derived, Hall	C₅×C₉×C₁₀₁ ⋊ C₂²
Hall	C₉×C₁₀₁ ⋊ C₂×D₅
Hall	C₅×C₁₀₁ ⋊ C₂×D₉
Sylow	C₁₀₁ ⋊ C₂×D₄₅
Hall	C₅×C₉ ⋊ C₂×D₁₀₁
Sylow	C₉ ⋊ C₂×D₅₀₅
Sylow	C₅ ⋊ C₂×D₉₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₅₄₅
commutator, cosocle, hypocenter	C₅×C₉×C₁₀₁	C₂²
Frattini	C₃	C₂×D₁₅₁₅
Fitting	C₂×C₅×C₉×C₁₀₁	C₂
socle	C₂×C₃×C₅×C₁₀₁	D₃
LCS	G ▷ C₅×C₉×C₁₀₁
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 9090

A minimal faithful permutation representation — 2 generators on 117 points:

(1,2)(3,4,5,6,7,8,9,10,11)(12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(4,11)(5,10)(6,9)(7,8)(13,16)(14,15)(18,117)(19,116)(20,115)(21,114)(22,113)(23,112)(24,111)(25,110)(26,109)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Conjugacy classes & character table

4548 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₁₈₀, C₂×D₁₃₆₃₅
maximal normal subgroup of	D₁₈₁₈₀
maximal quotient of	Dic₉₀₉₀, D₁₈₁₈₀, C₂×D₁₃₆₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18180, 42);

C2×D4545 #32911409