D₂₇₃₄₈ #72290658

order	54696
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₃₄₈

Splittings

Fitting	C₃×C₄×C₄₃×C₅₃ ⋊ C₂
Hall	C₃×C₄₃×C₅₃ ⋊ D₄
Hall	C₄₃×C₅₃ ⋊ D₁₂
Hall	C₃×C₅₃ ⋊ D₁₇₂
Hall	C₃×C₄₃ ⋊ D₂₁₂
Sylow	C₅₃ ⋊ D₅₁₆
Sylow	C₄₃ ⋊ D₆₃₆
Sylow	C₃ ⋊ D₉₁₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₈₃₇
commutator, socle, cosocle	C₂×C₃×C₄₃×C₅₃	C₂²
Fitting	C₃×C₄×C₄₃×C₅₃	C₂
hypercenter	C₄	D₆₈₃₇
hypocenter	C₃×C₄₃×C₅₃	D₄
LCS	G ▷ C₂×C₃×C₄₃×C₅₃ ▷ C₃×C₄₃×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 27348

A minimal faithful permutation representation — 2 generators on 103 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);(2,4)(6,7)(9,50)(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(52,103)(53,102)(54,101)(55,100)(56,99)(57,98)(58,97)(59,96)(60,95)(61,94)(62,93)(63,92)(64,91)(65,90)(66,89)(67,88)(68,87)(69,86)(70,85)(71,84)(72,83)(73,82)(74,81)(75,80)(76,79)(77,78)

Polycyclic presentation:

2^3,3,43,53|b^2=c,b^a=bc,d^a=d^2,e^a=e^42,f^a=f^52

Conjugacy classes & character table

13677 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, 2, 2, 3, 43, 53]);;
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.5^42);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27348 #72290658