C₂×D₃₂₈₅ #27190218

order	13140
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₅₇₀
direct product	C₂×D₃₂₈₅
unambiguous semidirect product	(C₂×C₅)⋊D₆₅₇
unambiguous semidirect product	(C₂×C₉)⋊D₃₆₅
unambiguous semidirect product	(C₂×C₇₃)⋊D₄₅

Splittings

Fitting	C₂×C₅×C₉×C₇₃ ⋊ C₂
derived, Hall	C₅×C₉×C₇₃ ⋊ C₂²
Hall	C₉×C₇₃ ⋊ C₂×D₅
Hall	C₅×C₇₃ ⋊ C₂×D₉
Sylow	C₇₃ ⋊ C₂×D₄₅
Hall	C₅×C₉ ⋊ C₂×D₇₃
Sylow	C₉ ⋊ C₂×D₃₆₅
Sylow	C₅ ⋊ C₂×D₆₅₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₈₅
commutator, cosocle, hypocenter	C₅×C₉×C₇₃	C₂²
Frattini	C₃	C₂×D₁₀₉₅
Fitting	C₂×C₅×C₉×C₇₃	C₂
socle	C₂×C₃×C₅×C₇₃	D₃
LCS	G ▷ C₅×C₉×C₇₃
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 6570

A minimal faithful permutation representation — 2 generators on 89 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);(4,11)(5,10)(6,9)(7,8)(13,16)(14,15)(18,89)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,81)(27,80)(28,79)(29,78)(30,77)(31,76)(32,75)(33,74)(34,73)(35,72)(36,71)(37,70)(38,69)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Conjugacy classes & character table

3288 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₃₁₄₀, C₂×D₉₈₅₅
maximal normal subgroup of	D₁₃₁₄₀
maximal quotient of	D₁₃₁₄₀, Dic₆₅₇₀, C₂×D₉₈₅₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 52);

C2×D3285 #27190218