D₁₆₃₈₅ #45658807

order	32770
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₃₈₅
unambiguous semidirect product	C₅⋊D₃₂₇₇
unambiguous semidirect product	C₂₉⋊D₅₆₅
unambiguous semidirect product	C₁₁₃⋊D₁₄₅

Splittings

Fitting, socle, derived, Hall	C₅×C₂₉×C₁₁₃ ⋊ C₂
Hall	C₂₉×C₁₁₃ ⋊ D₅
Hall	C₅×C₁₁₃ ⋊ D₂₉
Hall	C₅×C₂₉ ⋊ D₁₁₃
Sylow	C₁₁₃ ⋊ D₁₄₅
Sylow	C₂₉ ⋊ D₅₆₅
Sylow	C₅ ⋊ D₃₂₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₂₉×C₁₁₃	C₂
LCS	G ▷ C₅×C₂₉×C₁₁₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 147
transitive degree	μ_t(G) = 16385

A minimal faithful permutation representation — 2 generators on 147 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);(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)(36,147)(37,146)(38,145)(39,144)(40,143)(41,142)(42,141)(43,140)(44,139)(45,138)(46,137)(47,136)(48,135)(49,134)(50,133)(51,132)(52,131)(53,130)(54,129)(55,128)(56,127)(57,126)(58,125)(59,124)(60,123)(61,122)(62,121)(63,120)(64,119)(65,118)(66,117)(67,116)(68,115)(69,114)(70,113)(71,112)(72,111)(73,110)(74,109)(75,108)(76,107)(77,106)(78,105)(79,104)(80,103)(81,102)(82,101)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)

Polycyclic presentation:

2,5,29,113|b^a=b^4,c^a=c^28,d^a=d^112

Conjugacy classes & character table

8194 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32770, 8);

D16385 #45658807