Dic₈₆₉₇ #46852254

order	34788
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₆₉₇
unambiguous semidirect product	C₃⋊Dic₂₈₉₉
unambiguous semidirect product	C₂₂₃⋊Dic₃₉

Splittings

derived, Hall	C₃×C₁₃×C₂₂₃ ⋊ C₄
Hall	C₁₃×C₂₂₃ ⋊ Dic₃
Hall	C₃×C₂₂₃ ⋊ Dic₁₃
Sylow	C₂₂₃ ⋊ Dic₃₉
Hall	C₃×C₁₃ ⋊ Dic₂₂₃
Sylow	C₁₃ ⋊ Dic₆₆₉
Sylow	C₃ ⋊ Dic₂₈₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₆₉₇
commutator, hypocenter	C₃×C₁₃×C₂₂₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₁₃×C₂₂₃	C₂
LCS	G ▷ C₃×C₁₃×C₂₂₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 243
transitive degree	μ_t(G) = 34788

A minimal faithful permutation representation — 2 generators on 243 points:

(1,2)(3,4)(5,6,7)(8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243);(1,3,2,4)(6,7)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(22,243)(23,242)(24,241)(25,240)(26,239)(27,238)(28,237)(29,236)(30,235)(31,234)(32,233)(33,232)(34,231)(35,230)(36,229)(37,228)(38,227)(39,226)(40,225)(41,224)(42,223)(43,222)(44,221)(45,220)(46,219)(47,218)(48,217)(49,216)(50,215)(51,214)(52,213)(53,212)(54,211)(55,210)(56,209)(57,208)(58,207)(59,206)(60,205)(61,204)(62,203)(63,202)(64,201)(65,200)(66,199)(67,198)(68,197)(69,196)(70,195)(71,194)(72,193)(73,192)(74,191)(75,190)(76,189)(77,188)(78,187)(79,186)(80,185)(81,184)(82,183)(83,182)(84,181)(85,180)(86,179)(87,178)(88,177)(89,176)(90,175)(91,174)(92,173)(93,172)(94,171)(95,170)(96,169)(97,168)(98,167)(99,166)(100,165)(101,164)(102,163)(103,162)(104,161)(105,160)(106,159)(107,158)(108,157)(109,156)(110,155)(111,154)(112,153)(113,152)(114,151)(115,150)(116,149)(117,148)(118,147)(119,146)(120,145)(121,144)(122,143)(123,142)(124,141)(125,140)(126,139)(127,138)(128,137)(129,136)(130,135)(131,134)(132,133)

Polycyclic presentation:

2,3,13,223,2|a^2=e,b^a=b^2,c^a=c^12,d^a=d^222

Conjugacy classes & character table

8700 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34788, 12);

Dic8697 #46852254