Dic₈₇₆₃ #46856283

order	35052
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₇₆₃
unambiguous semidirect product	C₃⋊Dic₂₉₂₁
unambiguous semidirect product	C₂₃⋊Dic₃₈₁
unambiguous semidirect product	C₁₂₇⋊Dic₆₉

Splittings

derived, Hall	C₃×C₂₃×C₁₂₇ ⋊ C₄
Hall	C₂₃×C₁₂₇ ⋊ Dic₃
Hall	C₃×C₁₂₇ ⋊ Dic₂₃
Sylow	C₁₂₇ ⋊ Dic₆₉
Hall	C₃×C₂₃ ⋊ Dic₁₂₇
Sylow	C₂₃ ⋊ Dic₃₈₁
Sylow	C₃ ⋊ Dic₂₉₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₇₆₃
commutator, hypocenter	C₃×C₂₃×C₁₂₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₂₃×C₁₂₇	C₂
LCS	G ▷ C₃×C₂₃×C₁₂₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 157
transitive degree	μ_t(G) = 35052

A minimal faithful permutation representation — 2 generators on 157 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)(31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157);(1,3,2,4)(6,7)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(32,157)(33,156)(34,155)(35,154)(36,153)(37,152)(38,151)(39,150)(40,149)(41,148)(42,147)(43,146)(44,145)(45,144)(46,143)(47,142)(48,141)(49,140)(50,139)(51,138)(52,137)(53,136)(54,135)(55,134)(56,133)(57,132)(58,131)(59,130)(60,129)(61,128)(62,127)(63,126)(64,125)(65,124)(66,123)(67,122)(68,121)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(91,98)(92,97)(93,96)(94,95)

Polycyclic presentation:

2,3,23,127,2|a^2=e,b^a=b^2,c^a=c^22,d^a=d^126

Conjugacy classes & character table

8766 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35052, 9);

Dic8763 #46856283