D₁₅×C₁₃₉ #16861978

order	4170
isoclinism	isoclinic class of D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₁₅×C₁₃₉
unambiguous semidirect product	C₃⋊(D₅×C₁₃₉)
unambiguous semidirect product	C₅⋊(D₃×C₁₃₉)

Splittings

Fitting, socle, Hall	C₃×C₅×C₁₃₉ ⋊ C₂
derived, Hall	C₃×C₅ ⋊ C₂×C₁₃₉
Hall	C₅×C₁₃₉ ⋊ D₃
Hall	C₃×C₁₃₉ ⋊ D₅
Sylow	C₅ ⋊ D₃×C₁₃₉
Sylow	C₃ ⋊ D₅×C₁₃₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃₉	D₁₅
commutator, cosocle, hypocenter	C₃×C₅	C₂×C₁₃₉
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₁₃₉	C₂
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₁₃₉

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 2085

A minimal faithful permutation representation — 2 generators on 147 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(2,3)(5,8)(6,7)

Conjugacy classes & character table

1251 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₅×C₁₃₉, D₁₅×C₁₃₉⋊C₃, C₃⋊D₁₅×C₁₃₉, …
maximal normal subgroup of	D₁₅×C₁₃₉⋊C₃, C₃×D₁₅×C₁₃₉, C₅×D₁₅×C₁₃₉, …
maximal quotient of	D₄₅×C₁₃₉, C₃⋊D₁₅×C₁₃₉, C₃×D₁₅×C₁₃₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4170, 8);

D15×C139 #16861978