Dic₃×C₁₀₉ #13049994

order	1308
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₀₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	Dic₃×C₁₀₉
unambiguous semidirect product	C₃⋊(C₄×C₁₀₉)

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₁₀₉	D₃
commutator, hypocenter	C₃	C₄×C₁₀₉
Frattini	C₂	D₃×C₁₀₉
Fitting, socle, prime radical	C₂×C₃×C₁₀₉	C₂
cosocle	C₂×C₃	C₂×C₁₀₉
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₁₀₉

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);(1,3,2,4)(6,7)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	Dic₉×C₁₀₉, Dic₃×C₁₀₉⋊C₃, C₃×Dic₃×C₁₀₉, …
maximal normal subgroup of	Dic₃×C₁₀₉⋊C₃, C₃×Dic₃×C₁₀₉, C₁₁×Dic₃×C₁₀₉, …
maximal quotient of	Dic₉×C₁₀₉, C₃×Dic₃×C₁₀₉, C₃⋊Dic₃×C₁₀₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1308, 3);

Dic3×C109 #13049994