C₅×D₃×C₁₄₉ #17985016

order	4470
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₁₄₉ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 2235

A minimal faithful permutation representation — 2 generators on 157 points:

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

Conjugacy classes & character table

2235 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₅×D₉×C₁₄₉, C₅×C₃⋊D₃×C₁₄₉, C₃×C₅×D₃×C₁₄₉, …
maximal normal subgroup of	C₃×C₅×D₃×C₁₄₉, D₃×C₂₅×C₁₄₉, C₅²×D₃×C₁₄₉, …
maximal quotient of	C₅×D₉×C₁₄₉, C₅×C₃⋊D₃×C₁₄₉, C₃×C₅×D₃×C₁₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4470, 5);

C5×D3×C149 #17985016