D₅×C₁₆₉ #13133019

order	1690
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₆₉
unambiguous semidirect product	C₅⋊(C₂×C₁₆₉)

Splittings

Fitting, Hall	C₅×C₁₆₉ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₆₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₆₉	D₅
commutator, hypocenter	C₅	C₂×C₁₆₉
Frattini	C₁₃	D₅×C₁₃
Fitting	C₅×C₁₆₉	C₂
socle, cosocle	C₅×C₁₃	C₂×C₁₃
prime radical	D₅×C₁₃	C₁₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₆₉

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 174
transitive degree	μ_t(G) = 845

A minimal faithful permutation representation — 2 generators on 174 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174);(2,5)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₁₆₉, #16809979, D₅×D₁₆₉, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₁₆₉, #16809979, D₅×D₁₆₉, …
maximal quotient of	Dic₅×C₁₆₉, C₂×D₅×C₁₆₉, C₇×D₅×C₁₆₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1690, 1);

D5×C169 #13133019