D₈₅⋊D₅₉ #34095418

order	20060
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₈₅⋊D₅₉
unambiguous semidirect product	D₂₉₅⋊D₁₇
unambiguous semidirect product	D₁₀₀₃⋊D₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₇×C₅₉ ⋊ C₂²
Hall	C₁₇×C₅₉ ⋊ C₂×D₅
Hall	C₅×C₅₉ ⋊ C₂×D₁₇
Hall	C₅×C₁₇ ⋊ C₂×D₅₉
Sylow	C₅₉ ⋊ D₅×D₁₇
Sylow	C₁₇ ⋊ D₅×D₅₉
Sylow	C₅ ⋊ D₁₇×D₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₇×C₅₉	C₂²
LCS	G ▷ C₅×C₁₇×C₅₉
UCS	C₁

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 5015

A minimal faithful permutation representation — 2 generators on 81 points:

(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(78,79)(80,81);(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)(61,62)(63,64)(65,66)(67,68)(69,70)(71,72)(73,74)(75,76)(77,78,80,81,79)

Polycyclic presentation:

2^2,5,17,59|d^a=d^16,e^a=e^58,c^b=c^4,d^b=d^16

Conjugacy classes & character table

1316 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20060, 28);

D85⋊D59 #34095418