D₃×#16862006 #38736983

order	25074
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×#16862006
unambiguous semidirect product	C₃⋊(C₂×#16862006)

Splittings

Fitting, socle, derived	C₃×C₇×C₁₉₉ ⋊ C₂×C₃
Hall	C₃×#16862006 ⋊ C₂
Hall	C₇×C₁₉₉ ⋊ C₃×D₃
Sylow	C₁₉₉ ⋊ D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ D₃×C₁₉₉⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₉₉	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₉₉
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 209
transitive degree	μ_t(G) = 4179

A minimal faithful permutation representation — 2 generators on 209 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)(38,53,70)(41,56,67)(44,59,64)(47,62,61)(50,65,58)(52,68,55)(54,72,92)(57,75,89)(60,78,86)(63,81,83)(66,84,80)(69,87,77)(71,90,74)(73,94,117)(76,97,114)(79,100,111)(82,103,108)(85,106,105)(88,109,102)(91,112,99)(93,115,96)(95,119,137)(98,121,141)(101,124,138)(104,127,135)(107,130,132)(110,133,129)(113,136,126)(116,139,123)(118,142,120)(122,146,159)(125,148,163)(128,151,160)(131,153,158)(134,156,155)(140,161,150)(143,164,147)(144,162,145)(149,168,177)(152,170,179)(154,172,178)(157,174,176)(165,180,169)(166,181,167)(171,184,189)(173,185,191)(175,187,190)(182,192,183)(186,193,196)(188,194,197)(195,198,199)(201,202,203)(204,206,205)(208,209);(1,2,4,8,15,25,38,54,73,95,109,65,33,12,19,29,42,58,77,99,123,147,167,161,112,68,35,48,64,83,105,129,119,144,166,182,192,164,115,70,89,111,135,158,176,168,121,72,93,116,140,162,117,141,163,179,191,197,193,170,124,75,37,52,69,88,110,134,157,175,188,195,194,172,127,78,40,14,23,34,47,63,82,104,128,152,171,174,130,81,43,17,3,6,11,18,28,41,57,76,98,122,133,84,46,20,5,9,16,26,39,55,74,96,120,145,136,87,49,22,32,45,61,80,102,126,150,169,183,180,139,90,51,67,86,108,132,155,146,94,118,143,165,181,142,92,114,138,160,178,190,184,148,97,53,71,91,113,137,159,177,189,196,199,198,185,151,100,56,24,36,50,66,85,107,131,154,173,186,187,153,103,59,27,7,13,21,31,44,60,79,101,125,149,156,106,62,30,10)(200,201,202,204,203,205,206)(207,208,209)

Conjugacy classes & character table

1401 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25074, 65);

D3×#16862006 #38736983