C₃×C₅×D₃×C₁₄₉ #27211786

order	13410
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅×C₁₄₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×C₅×D₃×C₁₄₉
unambiguous semidirect product	C₃⋊(C₂×C₃×C₅×C₁₄₉)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₁₄₉ ⋊ C₂
derived	C₃ ⋊ C₂×C₃×C₅×C₁₄₉
Hall	C₃²×C₁₄₉ ⋊ C₂×C₅
Hall	C₃²×C₅ ⋊ C₂×C₁₄₉
Sylow	C₃² ⋊ C₂×C₅×C₁₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅×C₁₄₉	D₃
commutator, cosocle, hypocenter	C₃	C₂×C₃×C₅×C₁₄₉
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₁₄₉	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₃×C₅×C₁₄₉

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 160
transitive degree	μ_t(G) = 4470

A minimal faithful permutation representation — 2 generators on 160 points:

(1,2,3)(5,6)(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155)(156,157,158,159,160);(4,5,6)

Conjugacy classes & character table

6705 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅×D₃×D₁₄₉, C₃×D₃×D₅×C₁₄₉, C₃×D₃×D₇₄₅, …
maximal normal subgroup of	C₃×C₅×D₃×D₁₄₉, C₃×D₃×D₅×C₁₄₉, C₃×D₃×D₇₄₅, …
maximal quotient of	C₃×C₅×Dic₃×C₁₄₉, C₂×C₃×C₅×D₃×C₁₄₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13410, 17);

C3×C5×D3×C149 #27211786