D₅₁×C₁₃₁ #27211353

order	13362
isoclinism	isoclinic class of D₅₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅₁×C₁₃₁
unambiguous semidirect product	C₃⋊(D₁₇×C₁₃₁)
unambiguous semidirect product	C₁₇⋊(D₃×C₁₃₁)

Splittings

Fitting, socle, Hall	C₃×C₁₇×C₁₃₁ ⋊ C₂
derived, Hall	C₃×C₁₇ ⋊ C₂×C₁₃₁
Hall	C₁₇×C₁₃₁ ⋊ D₃
Hall	C₃×C₁₃₁ ⋊ D₁₇
Sylow	C₁₇ ⋊ D₃×C₁₃₁
Sylow	C₃ ⋊ D₁₇×C₁₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃₁	D₅₁
commutator, cosocle, hypocenter	C₃×C₁₇	C₂×C₁₃₁
Frattini	C₁	G
Fitting, socle	C₃×C₁₇×C₁₃₁	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁ ◁ C₁₃₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 151
transitive degree	μ_t(G) = 6681

A minimal faithful permutation representation — 2 generators on 151 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(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);(1,2,4,6,8,10,12,14,16,17,15,13,11,9,7,5,3)(18,19,20)

Conjugacy classes & character table

3537 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₅₁×D₁₃₁, D₃×D₁₇×C₁₃₁, D₅₁⋊D₁₃₁, …
maximal normal subgroup of	D₅₁×D₁₃₁, D₃×D₁₇×C₁₃₁, D₅₁⋊D₁₃₁, …
maximal quotient of	C₁₃₁×Dic₅₁, C₂×D₅₁×C₁₃₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13362, 7);

D51×C131 #27211353