D₁₈₅₆₅ #49150930

order	37130
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₈₅₆₅
unambiguous semidirect product	C₅⋊D₃₇₁₃
unambiguous semidirect product	C₄₇⋊D₃₉₅
unambiguous semidirect product	C₇₉⋊D₂₃₅

Splittings

Fitting, socle, derived, Hall	C₅×C₄₇×C₇₉ ⋊ C₂
Hall	C₄₇×C₇₉ ⋊ D₅
Hall	C₅×C₇₉ ⋊ D₄₇
Hall	C₅×C₄₇ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₂₃₅
Sylow	C₄₇ ⋊ D₃₉₅
Sylow	C₅ ⋊ D₃₇₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₄₇×C₇₉	C₂
LCS	G ▷ C₅×C₄₇×C₇₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 18565

A minimal faithful permutation representation — 2 generators on 131 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);(2,5)(3,4)(7,52)(8,51)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)

Polycyclic presentation:

2,5,47,79|b^a=b^4,c^a=c^46,d^a=d^78

Conjugacy classes & character table

9284 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37130, 8);

D18565 #49150930