Dic₇₈₆₉ #44538336

order	31476
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₆₉
unambiguous semidirect product	C₃⋊Dic₂₆₂₃
unambiguous semidirect product	C₄₃⋊Dic₁₈₃

Splittings

derived, Hall	C₃×C₄₃×C₆₁ ⋊ C₄
Hall	C₄₃×C₆₁ ⋊ Dic₃
Hall	C₃×C₆₁ ⋊ Dic₄₃
Hall	C₃×C₄₃ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₁₂₉
Sylow	C₄₃ ⋊ Dic₁₈₃
Sylow	C₃ ⋊ Dic₂₆₂₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₆₉
commutator, hypocenter	C₃×C₄₃×C₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₄₃×C₆₁	C₂
LCS	G ▷ C₃×C₄₃×C₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 31476

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)(6,7)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2,3,43,61,2|a^2=e,b^a=b^2,c^a=c^42,d^a=d^60

Conjugacy classes & character table

7872 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₇₃₈
maximal normal subgroup of	Dic₁₅₇₃₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31476, 12);

Dic7869 #44538336