D₁₇₈₈₅ #49088703

order	35770
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₇₈₈₅
unambiguous semidirect product	C₅⋊D₃₅₇₇
unambiguous semidirect product	C₄₉⋊D₃₆₅
unambiguous semidirect product	C₇₃⋊D₂₄₅

Splittings

Fitting, derived, Hall	C₅×C₄₉×C₇₃ ⋊ C₂
Hall	C₄₉×C₇₃ ⋊ D₅
Hall	C₅×C₇₃ ⋊ D₄₉
Hall	C₅×C₄₉ ⋊ D₇₃
Sylow	C₇₃ ⋊ D₂₄₅
Sylow	C₄₉ ⋊ D₃₆₅
Sylow	C₅ ⋊ D₃₅₇₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₄₉×C₇₃	C₂
Frattini	C₇	D₂₅₅₅
socle	C₅×C₇×C₇₃	D₇
LCS	G ▷ C₅×C₄₉×C₇₃
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 127
transitive degree	μ_t(G) = 17885

A minimal faithful permutation representation — 2 generators on 127 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);(2,5)(3,4)(7,54)(8,53)(9,52)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(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:

2,5,7,73,7|c^7=e^6,b^a=b^4,c^a=c^6e,d^a=d^72,e^a=e^6

Conjugacy classes & character table

8944 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35770, 8);

D17885 #49088703