D₅×C₁₅₁ #13094099

order	1510
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₅₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₅₁
unambiguous semidirect product	C₅⋊(C₂×C₁₅₁)

Splittings

Fitting, socle, Hall	C₅×C₁₅₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₅₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₅₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₅₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₅₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₅₁

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 156
transitive degree	μ_t(G) = 755

A minimal faithful permutation representation — 2 generators on 156 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);(2,5)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	AGL(1,𝔽₅)×C₁₅₁, C₁₅₁⋊AGL(1,𝔽₅), D₅×D₁₅₁, …
maximal normal subgroup of	AGL(1,𝔽₅)×C₁₅₁, C₁₅₁⋊AGL(1,𝔽₅), D₅×D₁₅₁, …
maximal quotient of	Dic₅×C₁₅₁, C₂×D₅×C₁₅₁, C₃×D₅×C₁₅₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1510, 3);

D5×C151 #13094099