D₂₀₃₈₅ #72274445

order	40770
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₂₀₃₈₅

Splittings

Fitting, derived, Hall	C₅×C₂₇×C₁₅₁ ⋊ C₂
Hall	C₂₇×C₁₅₁ ⋊ D₅
Hall	C₅×C₁₅₁ ⋊ D₂₇
Sylow	C₁₅₁ ⋊ D₁₃₅
Hall	C₅×C₂₇ ⋊ D₁₅₁
Sylow	C₂₇ ⋊ D₇₅₅
Sylow	C₅ ⋊ D₄₀₇₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₅×C₂₇×C₁₅₁	C₂
Frattini	C₉	D₂₂₆₅
socle	C₃×C₅×C₁₅₁	D₉
LCS	G ▷ C₅×C₂₇×C₁₅₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 183
transitive degree	μ_t(G) = 20385

A minimal faithful permutation representation — 2 generators on 183 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,32)(30,31)(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:

2,3,5,151,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^4,d^a=d^150,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

10194 conjugacy classes. The character table is too large to display.

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, 3, 5, 151, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^150);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D20385 #72274445