C₅×D₃×C₁₀₇ #15692014

order	3210
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₁₀₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₃×C₁₀₇
unambiguous semidirect product	C₃⋊(C₂×C₅×C₁₀₇)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₀₇ ⋊ C₂
derived, Sylow	C₃ ⋊ C₂×C₅×C₁₀₇
Hall	C₃×C₁₀₇ ⋊ C₂×C₅
Hall	C₃×C₅ ⋊ C₂×C₁₀₇

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₁₀₇	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₅×C₁₀₇
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₀₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₅×C₁₀₇

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 1605

A minimal faithful permutation representation — 2 generators on 115 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);(7,8)

Conjugacy classes & character table

1605 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃×C₂₅×C₁₀₇, C₅×D₁₅×C₁₀₇, C₅²×D₃×C₁₀₇, …
maximal normal subgroup of	D₃×C₂₅×C₁₀₇, C₅²×D₃×C₁₀₇, C₅×D₃×C₇×C₁₀₇, …
maximal quotient of	D₃×C₂₅×C₁₀₇, C₅×D₁₅×C₁₀₇, C₅²×D₃×C₁₀₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3210, 5);

C5×D3×C107 #15692014