D₁₃₅₃₂ #72235510

order	27064
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₃₅₃₂

Splittings

Fitting	C₄×C₁₇×C₁₉₉ ⋊ C₂
Hall	C₁₇×C₁₉₉ ⋊ D₄
Sylow	C₁₉₉ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₇₉₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₃₃₈₃
commutator, socle, cosocle	C₂×C₁₇×C₁₉₉	C₂²
Fitting	C₄×C₁₇×C₁₉₉	C₂
hypercenter	C₄	D₃₃₈₃
hypocenter	C₁₇×C₁₉₉	D₄
LCS	G ▷ C₂×C₁₇×C₁₉₉ ▷ C₁₇×C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 220
transitive degree	μ_t(G) = 13532

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

Polycyclic presentation:

2^3,17,199|b^2=c,b^a=bc,d^a=d^16,e^a=e^198

Conjugacy classes & character table

6769 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₇₀₆₄
maximal normal subgroup of	D₂₇₀₆₄
maximal quotient of	Dic₁₃₅₃₂, D₂₇₀₆₄

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 17, 199]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^16);;
gap> SetConjugate(coll, 5, 1, F.5^198);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D13532 #72235510