D₁₀₃₈₅ #35209352

order	20770
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₀₃₈₅
unambiguous semidirect product	C₅⋊D₂₀₇₇
unambiguous semidirect product	C₃₁⋊D₃₃₅
unambiguous semidirect product	C₆₇⋊D₁₅₅

Splittings

Fitting, socle, derived, Hall	C₅×C₃₁×C₆₇ ⋊ C₂
Hall	C₃₁×C₆₇ ⋊ D₅
Hall	C₅×C₆₇ ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₁₅₅
Sylow	C₃₁ ⋊ D₃₃₅
Sylow	C₅ ⋊ D₂₀₇₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₃₁×C₆₇	C₂
LCS	G ▷ C₅×C₃₁×C₆₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 10385

A minimal faithful permutation representation — 2 generators on 103 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);(2,5)(3,4)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2,5,31,67|b^a=b^4,c^a=c^30,d^a=d^66

Conjugacy classes & character table

5194 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20770, 9);

D10385 #35209352