D₁₈₄₁₅ #49145697

order	36830
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₄₁₅
unambiguous semidirect product	C₅⋊D₃₆₈₃
unambiguous semidirect product	C₂₉⋊D₆₃₅
unambiguous semidirect product	C₁₂₇⋊D₁₄₅

Splittings

Fitting, socle, derived, Hall	C₅×C₂₉×C₁₂₇ ⋊ C₂
Hall	C₂₉×C₁₂₇ ⋊ D₅
Hall	C₅×C₁₂₇ ⋊ D₂₉
Hall	C₅×C₂₉ ⋊ D₁₂₇
Sylow	C₁₂₇ ⋊ D₁₄₅
Sylow	C₂₉ ⋊ D₆₃₅
Sylow	C₅ ⋊ D₃₆₈₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₂₉×C₁₂₇	C₂
LCS	G ▷ C₅×C₂₉×C₁₂₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 161
transitive degree	μ_t(G) = 18415

A minimal faithful permutation representation — 2 generators on 161 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);(2,5)(3,4)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,135)(63,134)(64,133)(65,132)(66,131)(67,130)(68,129)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)

Polycyclic presentation:

2,5,29,127|b^a=b^4,c^a=c^28,d^a=d^126

Conjugacy classes & character table

9209 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36830, 8);

D18415 #49145697