D₃₀₈₂₅ #72296554

order	61650
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₈₂₅

Splittings

Fitting, derived, Hall	C₉×C₂₅×C₁₃₇ ⋊ C₂
Hall	C₂₅×C₁₃₇ ⋊ D₉
Hall	C₉×C₁₃₇ ⋊ D₂₅
Hall	C₉×C₂₅ ⋊ D₁₃₇
Sylow	C₁₃₇ ⋊ D₂₂₅
Sylow	C₂₅ ⋊ D₁₂₃₃
Sylow	C₉ ⋊ D₃₄₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₅×C₁₃₇	C₂
Frattini	C₃×C₅	D₂₀₅₅
socle	C₃×C₅×C₁₃₇	D₁₅
LCS	G ▷ C₉×C₂₅×C₁₃₇
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 171
transitive degree	μ_t(G) = 30825

A minimal faithful permutation representation — 2 generators on 171 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);(2,9)(3,8)(4,7)(5,6)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(36,171)(37,170)(38,169)(39,168)(40,167)(41,166)(42,165)(43,164)(44,163)(45,162)(46,161)(47,160)(48,159)(49,158)(50,157)(51,156)(52,155)(53,154)(54,153)(55,152)(56,151)(57,150)(58,149)(59,148)(60,147)(61,146)(62,145)(63,144)(64,143)(65,142)(66,141)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,126)(82,125)(83,124)(84,123)(85,122)(86,121)(87,120)(88,119)(89,118)(90,117)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)

Polycyclic presentation:

2,3,5,137,3,5|b^3=e,c^5=f,b^a=b^2e^2,c^a=c^4f^4,d^a=d^136,e^a=e^2,f^a=f^4

Conjugacy classes & character table

15414 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, 137, 3, 5]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2);;
gap> SetConjugate(coll, 3, 1, F.3^4*F.6^4);;
gap> SetConjugate(coll, 4, 1, F.4^136);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30825 #72296554