C₂×D₄₃₆₅ #31775516

order	17460
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₈₇₃₀
direct product	C₂×D₄₃₆₅
unambiguous semidirect product	(C₂×C₅)⋊D₈₇₃
unambiguous semidirect product	(C₂×C₉)⋊D₄₈₅
unambiguous semidirect product	(C₂×C₉₇)⋊D₄₅

Splittings

Fitting	C₂×C₅×C₉×C₉₇ ⋊ C₂
derived, Hall	C₅×C₉×C₉₇ ⋊ C₂²
Hall	C₉×C₉₇ ⋊ C₂×D₅
Hall	C₅×C₉₇ ⋊ C₂×D₉
Sylow	C₉₇ ⋊ C₂×D₄₅
Hall	C₅×C₉ ⋊ C₂×D₉₇
Sylow	C₉ ⋊ C₂×D₄₈₅
Sylow	C₅ ⋊ C₂×D₈₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₃₆₅
commutator, cosocle, hypocenter	C₅×C₉×C₉₇	C₂²
Frattini	C₃	C₂×D₁₄₅₅
Fitting	C₂×C₅×C₉×C₉₇	C₂
socle	C₂×C₃×C₅×C₉₇	D₃
LCS	G ▷ C₅×C₉×C₉₇
UCS	C₁ ◁ C₂

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 8730

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

Conjugacy classes & character table

4368 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₇₄₆₀, C₂×D₁₃₀₉₅
maximal normal subgroup of	D₁₇₄₆₀
maximal quotient of	Dic₈₇₃₀, D₁₇₄₆₀, C₂×D₁₃₀₉₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17460, 52);

C2×D4365 #31775516