Dic₈₇₆₉ #47975131

order	35076
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₇₆₉
unambiguous semidirect product	C₃⋊Dic₂₉₂₃
unambiguous semidirect product	C₇₉⋊Dic₁₁₁

Splittings

derived, Hall	C₃×C₃₇×C₇₉ ⋊ C₄
Hall	C₃₇×C₇₉ ⋊ Dic₃
Hall	C₃×C₇₉ ⋊ Dic₃₇
Hall	C₃×C₃₇ ⋊ Dic₇₉
Sylow	C₇₉ ⋊ Dic₁₁₁
Sylow	C₃₇ ⋊ Dic₂₃₇
Sylow	C₃ ⋊ Dic₂₉₂₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₇₆₉
commutator, hypocenter	C₃×C₃₇×C₇₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₃₇×C₇₉	C₂
LCS	G ▷ C₃×C₃₇×C₇₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 35076

A minimal faithful permutation representation — 2 generators on 123 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)(45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3,2,4)(6,7)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(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:

2,3,37,79,2|a^2=e,b^a=b^2,c^a=c^36,d^a=d^78

Conjugacy classes & character table

8772 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35076, 12);

Dic8769 #47975131