Dic₈₁₈₃ #45658596

order	32732
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₁₈₃
unambiguous semidirect product	C₄₉⋊Dic₁₆₇
unambiguous semidirect product	C₁₆₇⋊Dic₄₉

Splittings

derived, Hall	C₄₉×C₁₆₇ ⋊ C₄
Sylow	C₁₆₇ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₁₆₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₈₁₈₃
commutator, hypocenter	C₄₉×C₁₆₇	C₄
Frattini	C₂×C₇	D₁₁₆₉
Fitting, cosocle	C₂×C₄₉×C₁₆₇	C₂
socle, prime radical	C₂×C₇×C₁₆₇	D₇
LCS	G ▷ C₄₉×C₁₆₇
UCS	C₁ ◁ C₂

18 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 220
transitive degree	μ_t(G) = 32732

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

Polycyclic presentation:

2,7,167,2,7|a^2=d,b^7=e^6,b^a=b^6e,c^a=c^166,e^a=e^6

Conjugacy classes & character table

8186 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₃₆₆
maximal normal subgroup of	Dic₁₆₃₆₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32732, 4);

Dic8183 #45658596