C₇×C₁₀₉⋊C₃ #14556232

order	2289
isoclinism	isoclinic class of C₁₀₉⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₇×C₁₀₉⋊C₃
unambiguous semidirect product	C₁₀₉⋊(C₃×C₇)

Splittings

Fitting, socle, Hall	C₇×C₁₀₉ ⋊ C₃
derived, Sylow	C₁₀₉ ⋊ C₃×C₇

Subgroups

	subgroup	G/H
center, hypercenter	C₇	C₁₀₉⋊C₃
commutator, cosocle, hypocenter	C₁₀₉	C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇×C₁₀₉	C₃
LCS	G ▷ C₁₀₉
UCS	C₁ ◁ C₇

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 116
transitive degree	μ_t(G) = 763

A minimal faithful permutation representation — 2 generators on 116 points:

(1,2,3,4,5,6,7)(9,10,12)(11,14,19)(13,17,16)(15,21,29)(18,24,26)(20,27,23)(22,31,42)(25,34,39)(28,37,36)(30,40,33)(32,44,58)(35,47,55)(38,50,52)(41,53,49)(43,56,46)(45,60,76)(48,62,73)(51,65,70)(54,68,67)(57,71,64)(59,74,61)(63,77,90)(66,79,87)(69,82,84)(72,85,81)(75,88,78)(80,91,101)(83,93,98)(86,96,95)(89,99,92)(94,102,109)(97,104,106)(100,107,103)(105,110,114)(108,112,111)(113,115,116);(8,9,11,15,22,32,45,58,73,87,98,106,111,110,93,65,34,14,20,28,38,51,66,80,88,56,29,39,52,67,81,92,91,62,31,43,57,72,86,97,105,107,85,53,27,12,16,23,33,46,61,60,76,90,101,109,114,116,115,104,82,50,24,10,13,18,25,35,48,63,74,42,55,70,84,95,103,102,79,47,21,30,41,54,69,83,94,99,71,40,19,26,36,49,64,78,77,44,59,75,89,100,108,113,112,96,68,37,17)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₇×C₁₀₉⋊C₃, C₇×D₁₀₉⋊C₃, D₇×C₁₀₉⋊C₃, …
maximal normal subgroup of	C₂×C₇×C₁₀₉⋊C₃, C₇×D₁₀₉⋊C₃, D₇×C₁₀₉⋊C₃, …
maximal quotient of	C₂×C₇×C₁₀₉⋊C₃, C₅×C₇×C₁₀₉⋊C₃, C₄₉×C₁₀₉⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2289, 2);

C7×C109⋊C3 #14556232