D₁₁₇₆₄ #72223958

order	23528
automorphism	Aut(G) order 64749056
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₇₆₄

Splittings

Fitting	C₄×C₁₇×C₁₇₃ ⋊ C₂
Hall	C₁₇×C₁₇₃ ⋊ D₄
Sylow	C₁₇₃ ⋊ D₆₈
Sylow	C₁₇ ⋊ D₆₉₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₄₁
commutator, socle, cosocle	C₂×C₁₇×C₁₇₃	C₂²
Fitting	C₄×C₁₇×C₁₇₃	C₂
hypercenter	C₄	D₂₉₄₁
hypocenter	C₁₇×C₁₇₃	D₄
LCS	G ▷ C₂×C₁₇×C₁₇₃ ▷ C₁₇×C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 194
transitive degree	μ_t(G) = 11764

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

Polycyclic presentation:

2^3,17,173|b^2=c,b^a=bc,d^a=d^16,e^a=e^172

Conjugacy classes & character table

5885 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₅₂₈
maximal normal subgroup of	D₂₃₅₂₈
maximal quotient of	Dic₁₁₇₆₄, D₂₃₅₂₈

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, 173]);;
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.5^172);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11764 #72223958