Dic₇₈₈₁ #44538607

order	31524
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₈₁
unambiguous semidirect product	C₃⋊Dic₂₆₂₇
unambiguous semidirect product	C₇₁⋊Dic₁₁₁

Splittings

derived, Hall	C₃×C₃₇×C₇₁ ⋊ C₄
Hall	C₃₇×C₇₁ ⋊ Dic₃
Hall	C₃×C₇₁ ⋊ Dic₃₇
Hall	C₃×C₃₇ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₁₁₁
Sylow	C₃₇ ⋊ Dic₂₁₃
Sylow	C₃ ⋊ Dic₂₆₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₈₁
commutator, hypocenter	C₃×C₃₇×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₃₇×C₇₁	C₂
LCS	G ▷ C₃×C₃₇×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 31524

A minimal faithful permutation representation — 2 generators on 115 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);(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)(46,115)(47,114)(48,113)(49,112)(50,111)(51,110)(52,109)(53,108)(54,107)(55,106)(56,105)(57,104)(58,103)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,87)(75,86)(76,85)(77,84)(78,83)(79,82)(80,81)

Polycyclic presentation:

2,3,37,71,2|a^2=e,b^a=b^2,c^a=c^36,d^a=d^70

Conjugacy classes & character table

7884 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₇₆₂
maximal normal subgroup of	Dic₁₅₇₆₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31524, 9);

Dic7881 #44538607