D₁₆₈₂₇ #46777823

order	33654
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₈₂₇
unambiguous semidirect product	C₃⋊D₅₆₀₉
unambiguous semidirect product	C₇₁⋊D₂₃₇
unambiguous semidirect product	C₇₉⋊D₂₁₃

Splittings

Fitting, socle, derived, Hall	C₃×C₇₁×C₇₉ ⋊ C₂
Hall	C₇₁×C₇₉ ⋊ D₃
Hall	C₃×C₇₉ ⋊ D₇₁
Hall	C₃×C₇₁ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₂₁₃
Sylow	C₇₁ ⋊ D₂₃₇
Sylow	C₃ ⋊ D₅₆₀₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇₁×C₇₉	C₂
LCS	G ▷ C₃×C₇₁×C₇₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 153
transitive degree	μ_t(G) = 16827

A minimal faithful permutation representation — 2 generators on 153 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);(2,3)(5,74)(6,73)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,54)(26,53)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(76,153)(77,152)(78,151)(79,150)(80,149)(81,148)(82,147)(83,146)(84,145)(85,144)(86,143)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)

Polycyclic presentation:

2,3,71,79|b^a=b^2,c^a=c^70,d^a=d^78

Conjugacy classes & character table

8415 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33654, 9);

D16827 #46777823