Dic₈₃₁₉ #45685408

order	33276
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₃₁₉
unambiguous semidirect product	C₃⋊Dic₂₇₇₃
unambiguous semidirect product	C₄₇⋊Dic₁₇₇
unambiguous semidirect product	C₅₉⋊Dic₁₄₁

Splittings

derived, Hall	C₃×C₄₇×C₅₉ ⋊ C₄
Hall	C₄₇×C₅₉ ⋊ Dic₃
Hall	C₃×C₅₉ ⋊ Dic₄₇
Hall	C₃×C₄₇ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₄₁
Sylow	C₄₇ ⋊ Dic₁₇₇
Sylow	C₃ ⋊ Dic₂₇₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₃₁₉
commutator, hypocenter	C₃×C₄₇×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₇×C₅₉	C₂
LCS	G ▷ C₃×C₄₇×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 33276

A minimal faithful permutation representation — 2 generators on 113 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);(1,3,2,4)(6,7)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2,3,47,59,2|a^2=e,b^a=b^2,c^a=c^46,d^a=d^58

Conjugacy classes & character table

8322 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33276, 8);

Dic8319 #45685408