D₃×#16860996 #37566555

order	24318
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₃×#16860996
unambiguous semidirect product	C₃⋊(C₂×#16860996)

Splittings

Fitting, socle, derived	C₃×C₇×C₁₉₃ ⋊ C₂×C₃
Hall	C₃×#16860996 ⋊ C₂
Hall	C₇×C₁₉₃ ⋊ C₃×D₃
Sylow	C₁₉₃ ⋊ D₃×ΓL(1,𝔽₈)
Sylow	C₇ ⋊ D₃×C₁₉₃⋊C₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₉₃	C₂×C₃
LCS	G ▷ C₃×C₇×C₁₉₃
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 203
transitive degree	μ_t(G) = 4053

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

Conjugacy classes & character table

1359 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24318, 54);

D3×#16860996 #37566555