Dic₉₀₈₅ #49138144

order	36340
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₀₈₅
unambiguous semidirect product	C₂₃⋊Dic₃₉₅
unambiguous semidirect product	C₇₉⋊Dic₁₁₅

Splittings

derived, Hall	C₅×C₂₃×C₇₉ ⋊ C₄
Hall	C₂₃×C₇₉ ⋊ Dic₅
Hall	C₅×C₇₉ ⋊ Dic₂₃
Hall	C₅×C₂₃ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₁₅
Sylow	C₂₃ ⋊ Dic₃₉₅
Sylow	C₅ ⋊ Dic₁₈₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₉₀₈₅
commutator, hypocenter	C₅×C₂₃×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₂₃×C₇₉	C₂
LCS	G ▷ C₅×C₂₃×C₇₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 36340

A minimal faithful permutation representation — 2 generators on 111 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);(1,3,2,4)(6,9)(7,8)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2,5,23,79,2|a^2=e,b^a=b^4,c^a=c^22,d^a=d^78

Conjugacy classes & character table

9088 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36340, 8);

Dic9085 #49138144