D₃×#16862571 #38784131

order	25506
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×#16862571
unambiguous semidirect product	C₃⋊(C₂×#16862571)

Splittings

Fitting, socle, derived	C₃×C₁₃×C₁₀₉ ⋊ C₂×C₃
Hall	C₃×#16862571 ⋊ C₂
Hall	C₁₃×C₁₀₉ ⋊ C₃×D₃
Sylow	C₁₀₉ ⋊ D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ D₃×C₁₀₉⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₃×C₁₀₉	C₂×C₃
LCS	G ▷ C₃×C₁₃×C₁₀₉
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 4251

A minimal faithful permutation representation — 2 generators on 125 points:

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

Conjugacy classes & character table

1425 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25506, 66);

D3×#16862571 #38784131