D₃×#17984521 #40998129

order	26586
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×#17984521
unambiguous semidirect product	C₃⋊(C₂×#17984521)

Splittings

Fitting, socle, derived	C₃×C₇×C₂₁₁ ⋊ C₂×C₃
Hall	C₃×#17984521 ⋊ C₂
Hall	C₇×C₂₁₁ ⋊ C₃×D₃
Sylow	C₂₁₁ ⋊ D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ D₃×C₂₁₁⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₂₁₁	C₂×C₃
LCS	G ▷ C₃×C₇×C₂₁₁
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 221
transitive degree	μ_t(G) = 4431

A minimal faithful permutation representation — 2 generators on 221 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)(95,119,144)(98,122,141)(101,125,138)(104,128,135)(107,120,132)(110,133,130)(113,136,127)(116,139,124)(118,142,121)(123,147,168)(126,150,165)(129,153,162)(131,148,159)(134,158,157)(137,160,155)(140,163,152)(143,166,149)(145,169,146)(151,172,187)(154,175,184)(156,173,181)(161,180,179)(164,182,177)(167,185,174)(170,188,171)(176,191,200)(178,192,197)(183,196,195)(186,198,193)(189,201,190)(194,204,207)(199,206,205)(202,208,203)(209,211,210)(213,214,215)(216,218,217)(220,221);(1,2,4,8,15,25,38,54,73,95,120,81,43,17,3,6,11,18,28,41,57,76,98,123,148,103,59,27,7,13,21,31,44,60,79,101,126,151,173,128,78,40,14,23,34,47,63,82,104,129,154,176,192,153,100,56,24,36,50,66,85,107,131,156,178,194,204,175,125,75,37,52,69,88,110,134,144,168,187,200,207,191,150,97,53,71,91,113,137,161,169,117,141,165,184,197,172,122,72,93,116,140,164,183,188,142,92,114,138,162,181,147,94,118,143,167,186,199,201,166,115,70,89,111,135,159,119,145,170,189,202,209,208,185,139,90,51,67,86,108,132,157,179,195,205,210,211,198,163,112,68,35,48,64,83,105,130,155,177,193,203,206,182,136,87,49,22,32,45,61,80,102,127,152,174,190,196,160,109,65,33,12,19,29,42,58,77,99,124,149,171,180,133,84,46,20,5,9,16,26,39,55,74,96,121,146,158,106,62,30,10)(212,213,214,216,215,217,218)(219,220,221)

Conjugacy classes & character table

1485 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26586, 69);

D3×#17984521 #40998129