Dic₈₆₁₉ #46828131

order	34476
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₆₁₉
unambiguous semidirect product	C₃⋊Dic₂₈₇₃

Splittings

derived, Hall	C₃×C₁₇×C₁₆₉ ⋊ C₄
Hall	C₁₇×C₁₆₉ ⋊ Dic₃
Hall	C₃×C₁₆₉ ⋊ Dic₁₇
Sylow	C₁₆₉ ⋊ Dic₅₁
Hall	C₃×C₁₇ ⋊ Dic₁₆₉
Sylow	C₁₇ ⋊ Dic₅₀₇
Sylow	C₃ ⋊ Dic₂₈₇₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₆₁₉
commutator, hypocenter	C₃×C₁₇×C₁₆₉	C₄
Frattini	C₂×C₁₃	D₆₆₃
Fitting, cosocle	C₂×C₃×C₁₇×C₁₆₉	C₂
socle, prime radical	C₂×C₃×C₁₃×C₁₇	D₁₃
LCS	G ▷ C₃×C₁₇×C₁₆₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 193
transitive degree	μ_t(G) = 34476

A minimal faithful permutation representation — 2 generators on 193 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193);(1,3,2,4)(6,7)(9,176)(10,175)(11,174)(12,173)(13,172)(14,171)(15,170)(16,169)(17,168)(18,167)(19,166)(20,165)(21,164)(22,163)(23,162)(24,161)(25,160)(26,159)(27,158)(28,157)(29,156)(30,155)(31,154)(32,153)(33,152)(34,151)(35,150)(36,149)(37,148)(38,147)(39,146)(40,145)(41,144)(42,143)(43,142)(44,141)(45,140)(46,139)(47,138)(48,137)(49,136)(50,135)(51,134)(52,133)(53,132)(54,131)(55,130)(56,129)(57,128)(58,127)(59,126)(60,125)(61,124)(62,123)(63,122)(64,121)(65,120)(66,119)(67,118)(68,117)(69,116)(70,115)(71,114)(72,113)(73,112)(74,111)(75,110)(76,109)(77,108)(78,107)(79,106)(80,105)(81,104)(82,103)(83,102)(84,101)(85,100)(86,99)(87,98)(88,97)(89,96)(90,95)(91,94)(92,93)(178,193)(179,192)(180,191)(181,190)(182,189)(183,188)(184,187)(185,186)

Polycyclic presentation:

2,3,13,17,2,13|a^2=e,c^13=f^12,b^a=b^2,c^a=c^12f,d^a=d^16,f^a=f^12

Conjugacy classes & character table

8622 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34476, 12);

Dic8619 #46828131