Dic₈₅₅₅ #46803754

order	34220
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₅₅₅
unambiguous semidirect product	C₅₉⋊Dic₁₄₅

Splittings

derived, Hall	C₅×C₂₉×C₅₉ ⋊ C₄
Hall	C₂₉×C₅₉ ⋊ Dic₅
Hall	C₅×C₅₉ ⋊ Dic₂₉
Hall	C₅×C₂₉ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₄₅
Sylow	C₂₉ ⋊ Dic₂₉₅
Sylow	C₅ ⋊ Dic₁₇₁₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₅₅₅
commutator, hypocenter	C₅×C₂₉×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₉×C₅₉	C₂
LCS	G ▷ C₅×C₂₉×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 34220

A minimal faithful permutation representation — 2 generators on 97 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);(1,3,2,4)(6,9)(7,8)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2,5,29,59,2|a^2=e,b^a=b^4,c^a=c^28,d^a=d^58

Conjugacy classes & character table

8558 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34220, 9);

Dic8555 #46803754