D₁₅₆₃₅ #44530787

order	31270
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₆₃₅
unambiguous semidirect product	C₅⋊D₃₁₂₇
unambiguous semidirect product	C₅₃⋊D₂₉₅
unambiguous semidirect product	C₅₉⋊D₂₆₅

Splittings

Fitting, socle, derived, Hall	C₅×C₅₃×C₅₉ ⋊ C₂
Hall	C₅₃×C₅₉ ⋊ D₅
Hall	C₅×C₅₉ ⋊ D₅₃
Hall	C₅×C₅₃ ⋊ D₅₉
Sylow	C₅₉ ⋊ D₂₆₅
Sylow	C₅₃ ⋊ D₂₉₅
Sylow	C₅ ⋊ D₃₁₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₅₃×C₅₉	C₂
LCS	G ▷ C₅×C₅₃×C₅₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 15635

A minimal faithful permutation representation — 2 generators on 117 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);(2,5)(3,4)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Polycyclic presentation:

2,5,53,59|b^a=b^4,c^a=c^52,d^a=d^58

Conjugacy classes & character table

7819 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₆₃₅
maximal normal subgroup of	C₂×D₁₅₆₃₅
maximal quotient of	Dic₁₅₆₃₅, C₂×D₁₅₆₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31270, 8);

D15635 #44530787