D₁₆₅₈₃ #45684254

order	33166
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₅₈₃
unambiguous semidirect product	C₇⋊D₂₃₆₉
unambiguous semidirect product	C₂₃⋊D₇₂₁
unambiguous semidirect product	C₁₀₃⋊D₁₆₁

Splittings

Fitting, socle, derived, Hall	C₇×C₂₃×C₁₀₃ ⋊ C₂
Hall	C₂₃×C₁₀₃ ⋊ D₇
Hall	C₇×C₁₀₃ ⋊ D₂₃
Hall	C₇×C₂₃ ⋊ D₁₀₃
Sylow	C₁₀₃ ⋊ D₁₆₁
Sylow	C₂₃ ⋊ D₇₂₁
Sylow	C₇ ⋊ D₂₃₆₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₃×C₁₀₃	C₂
LCS	G ▷ C₇×C₂₃×C₁₀₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 133
transitive degree	μ_t(G) = 16583

A minimal faithful permutation representation — 2 generators on 133 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);(2,7)(3,6)(4,5)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(32,133)(33,132)(34,131)(35,130)(36,129)(37,128)(38,127)(39,126)(40,125)(41,124)(42,123)(43,122)(44,121)(45,120)(46,119)(47,118)(48,117)(49,116)(50,115)(51,114)(52,113)(53,112)(54,111)(55,110)(56,109)(57,108)(58,107)(59,106)(60,105)(61,104)(62,103)(63,102)(64,101)(65,100)(66,99)(67,98)(68,97)(69,96)(70,95)(71,94)(72,93)(73,92)(74,91)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)

Polycyclic presentation:

2,7,23,103|b^a=b^6,c^a=c^22,d^a=d^102

Conjugacy classes & character table

8293 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33166, 8);

D16583 #45684254