D₂₃₆₆₈ #72282425

order	47336
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₃₆₆₈

Splittings

Fitting	C₄×C₆₁×C₉₇ ⋊ C₂
Hall	C₆₁×C₉₇ ⋊ D₄
Sylow	C₉₇ ⋊ D₂₄₄
Sylow	C₆₁ ⋊ D₃₈₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅₉₁₇
commutator, socle, cosocle	C₂×C₆₁×C₉₇	C₂²
Fitting	C₄×C₆₁×C₉₇	C₂
hypercenter	C₄	D₅₉₁₇
hypocenter	C₆₁×C₉₇	D₄
LCS	G ▷ C₂×C₆₁×C₉₇ ▷ C₆₁×C₉₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 162
transitive degree	μ_t(G) = 23668

A minimal faithful permutation representation — 2 generators on 162 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);(2,4)(6,65)(7,64)(8,63)(9,62)(10,61)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,50)(22,49)(23,48)(24,47)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(67,162)(68,161)(69,160)(70,159)(71,158)(72,157)(73,156)(74,155)(75,154)(76,153)(77,152)(78,151)(79,150)(80,149)(81,148)(82,147)(83,146)(84,145)(85,144)(86,143)(87,142)(88,141)(89,140)(90,139)(91,138)(92,137)(93,136)(94,135)(95,134)(96,133)(97,132)(98,131)(99,130)(100,129)(101,128)(102,127)(103,126)(104,125)(105,124)(106,123)(107,122)(108,121)(109,120)(110,119)(111,118)(112,117)(113,116)(114,115)

Polycyclic presentation:

2^3,61,97|b^2=c,b^a=bc,d^a=d^60,e^a=e^96

Conjugacy classes & character table

11837 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 61, 97]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^60);;
gap> SetConjugate(coll, 5, 1, F.5^96);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D23668 #72282425