D₁₁₈₂₀ #72223972

order	23640
automorphism	Aut(G) order 37067520
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₈₂₀

Splittings

Fitting	C₃×C₄×C₅×C₁₉₇ ⋊ C₂
Hall	C₃×C₅×C₁₉₇ ⋊ D₄
Hall	C₅×C₁₉₇ ⋊ D₁₂
Hall	C₃×C₁₉₇ ⋊ D₂₀
Sylow	C₁₉₇ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₇₈₈
Sylow	C₅ ⋊ D₂₃₆₄
Sylow	C₃ ⋊ D₃₉₄₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₅₅
commutator, socle, cosocle	C₂×C₃×C₅×C₁₉₇	C₂²
Fitting	C₃×C₄×C₅×C₁₉₇	C₂
hypercenter	C₄	D₂₉₅₅
hypocenter	C₃×C₅×C₁₉₇	D₄
LCS	G ▷ C₂×C₃×C₅×C₁₉₇ ▷ C₃×C₅×C₁₉₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 209
transitive degree	μ_t(G) = 11820

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

Polycyclic presentation:

2^3,3,5,197|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^196

Conjugacy classes & character table

5913 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₆₄₀
maximal normal subgroup of	D₂₃₆₄₀
maximal quotient of	Dic₁₁₈₂₀, D₂₃₆₄₀

GAP

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

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 197]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^196);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11820 #72223972