C₃×Dic₁₀₉ #13049995

order	1308
isoclinism	isoclinic class of D₁₀₉
automorphism	?
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×Dic₁₀₉

Splittings

derived, Sylow	C₁₀₉ ⋊ C₃×C₄
Hall	C₃×C₁₀₉ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₀₉
commutator, hypocenter	C₁₀₉	C₃×C₄
Frattini	C₂	C₃×D₁₀₉
Fitting, socle, prime radical	C₂×C₃×C₁₀₉	C₂
cosocle	C₂×C₁₀₉	C₂×C₃
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₃

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 1308

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#16840975, C₉×Dic₁₀₉, C₃×Dic₁₀₉⋊C₃, …
maximal normal subgroup of	#16840975, C₉×Dic₁₀₉, C₃×Dic₁₀₉⋊C₃, …
maximal quotient of	C₉×Dic₁₀₉, C₃²×Dic₁₀₉, C₃×Dic₃₂₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1308, 4);

C3×Dic109 #13049995