Dic₇₈₉₇ #44539104

order	31588
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₇₈₉₇

Splittings

derived, Hall	C₅₃×C₁₄₉ ⋊ C₄
Sylow	C₁₄₉ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₉₇
commutator, hypocenter	C₅₃×C₁₄₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅₃×C₁₄₉	C₂
LCS	G ▷ C₅₃×C₁₄₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 206
transitive degree	μ_t(G) = 31588

A minimal faithful permutation representation — 2 generators on 206 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);(1,3,2,4)(6,57)(7,56)(8,55)(9,54)(10,53)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(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:

2,53,149,2|a^2=d,b^a=b^52,c^a=c^148

Conjugacy classes & character table

7900 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₇₉₄
maximal normal subgroup of	Dic₁₅₇₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31588, 4);

Dic7897 #44539104