D₁₇₅₄₉ #47975329

order	35098
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₅₄₉
unambiguous semidirect product	C₇⋊D₂₅₀₇
unambiguous semidirect product	C₂₃⋊D₇₆₃
unambiguous semidirect product	C₁₀₉⋊D₁₆₁

Splittings

Fitting, socle, derived, Hall	C₇×C₂₃×C₁₀₉ ⋊ C₂
Hall	C₂₃×C₁₀₉ ⋊ D₇
Hall	C₇×C₁₀₉ ⋊ D₂₃
Hall	C₇×C₂₃ ⋊ D₁₀₉
Sylow	C₁₀₉ ⋊ D₁₆₁
Sylow	C₂₃ ⋊ D₇₆₃
Sylow	C₇ ⋊ D₂₅₀₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₂₃×C₁₀₉	C₂
LCS	G ▷ C₇×C₂₃×C₁₀₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 139
transitive degree	μ_t(G) = 17549

A minimal faithful permutation representation — 2 generators on 139 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);(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)(32,139)(33,138)(34,137)(35,136)(36,135)(37,134)(38,133)(39,132)(40,131)(41,130)(42,129)(43,128)(44,127)(45,126)(46,125)(47,124)(48,123)(49,122)(50,121)(51,120)(52,119)(53,118)(54,117)(55,116)(56,115)(57,114)(58,113)(59,112)(60,111)(61,110)(62,109)(63,108)(64,107)(65,106)(66,105)(67,104)(68,103)(69,102)(70,101)(71,100)(72,99)(73,98)(74,97)(75,96)(76,95)(77,94)(78,93)(79,92)(80,91)(81,90)(82,89)(83,88)(84,87)(85,86)

Polycyclic presentation:

2,7,23,109|b^a=b^6,c^a=c^22,d^a=d^108

Conjugacy classes & character table

8776 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35098, 8);

D17549 #47975329