C₃×D₆₁₇₉ #49150487

order	37074
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₆₁₇₉
unambiguous semidirect product	C₁₆₇⋊(C₃×D₃₇)

Splittings

Fitting, socle, Hall	C₃×C₃₇×C₁₆₇ ⋊ C₂
derived, Hall	C₃₇×C₁₆₇ ⋊ C₂×C₃
Hall	C₃×C₁₆₇ ⋊ D₃₇
Sylow	C₁₆₇ ⋊ C₃×D₃₇
Hall	C₃×C₃₇ ⋊ D₁₆₇
Sylow	C₃₇ ⋊ C₃×D₁₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₆₁₇₉
commutator, cosocle, hypocenter	C₃₇×C₁₆₇	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₃₇×C₁₆₇	C₂
LCS	G ▷ C₃₇×C₁₆₇
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 207
transitive degree	μ_t(G) = 18537

A minimal faithful permutation representation — 2 generators on 207 points:

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

Conjugacy classes & character table

9273 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37074, 5);

C3×D6179 #49150487