C₂×D₃₂₇ #13050008

order	1308
isoclinism	isoclinic class of D₃₂₇
automorphism	Aut(G) order 141264
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₆₅₄
direct product	C₂×D₃₂₇
unambiguous semidirect product	(C₂×C₃)⋊D₁₀₉
unambiguous semidirect product	(C₂×C₁₀₉)⋊D₃

Splittings

Fitting, socle	C₂×C₃×C₁₀₉ ⋊ C₂
derived, Hall	C₃×C₁₀₉ ⋊ C₂²
Sylow	C₁₀₉ ⋊ C₂×D₃
Sylow	C₃ ⋊ C₂×D₁₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₂₇
commutator, cosocle, hypocenter	C₃×C₁₀₉	C₂²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₀₉	C₂
LCS	G ▷ C₃×C₁₀₉
UCS	C₁ ◁ C₂

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 114
transitive degree	μ_t(G) = 654

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×D₉₈₁, C₂×C₃⋊(D₁₀₉⋊C₃), D₃×D₃₂₇, …
maximal normal subgroup of	C₂×C₃⋊(D₁₀₉⋊C₃), C₂×C₃×D₃₂₇, C₂×C₁₁×D₃₂₇, …
maximal quotient of	C₂×D₉₈₁, D₃×D₃₂₇, C₂×C₃⋊D₃₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1308, 17);

C2×D327 #13050008