C₃×D₅₅₁₃ #45683446

order	33078
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅₅₁₃
unambiguous semidirect product	C₁₄₉⋊(C₃×D₃₇)

Splittings

Fitting, socle, Hall	C₃×C₃₇×C₁₄₉ ⋊ C₂
derived, Hall	C₃₇×C₁₄₉ ⋊ C₂×C₃
Hall	C₃×C₁₄₉ ⋊ D₃₇
Sylow	C₁₄₉ ⋊ C₃×D₃₇
Hall	C₃×C₃₇ ⋊ D₁₄₉
Sylow	C₃₇ ⋊ C₃×D₁₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₅₅₁₃
commutator, cosocle, hypocenter	C₃₇×C₁₄₉	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₃₇×C₁₄₉	C₂
LCS	G ▷ C₃₇×C₁₄₉
UCS	C₁ ◁ C₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 189
transitive degree	μ_t(G) = 16539

A minimal faithful permutation representation — 2 generators on 189 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);(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)(42,189)(43,188)(44,187)(45,186)(46,185)(47,184)(48,183)(49,182)(50,181)(51,180)(52,179)(53,178)(54,177)(55,176)(56,175)(57,174)(58,173)(59,172)(60,171)(61,170)(62,169)(63,168)(64,167)(65,166)(66,165)(67,164)(68,163)(69,162)(70,161)(71,160)(72,159)(73,158)(74,157)(75,156)(76,155)(77,154)(78,153)(79,152)(80,151)(81,150)(82,149)(83,148)(84,147)(85,146)(86,145)(87,144)(88,143)(89,142)(90,141)(91,140)(92,139)(93,138)(94,137)(95,136)(96,135)(97,134)(98,133)(99,132)(100,131)(101,130)(102,129)(103,128)(104,127)(105,126)(106,125)(107,124)(108,123)(109,122)(110,121)(111,120)(112,119)(113,118)(114,117)(115,116)

Conjugacy classes & character table

8274 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33078, 6);

C3×D5513 #45683446