D₃₂₄₁₂ #72302122

order	64824
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₄₁₂

Splittings

Fitting	C₃×C₄×C₃₇×C₇₃ ⋊ C₂
Hall	C₃×C₃₇×C₇₃ ⋊ D₄
Hall	C₃₇×C₇₃ ⋊ D₁₂
Hall	C₃×C₇₃ ⋊ D₁₄₈
Hall	C₃×C₃₇ ⋊ D₂₉₂
Sylow	C₇₃ ⋊ D₄₄₄
Sylow	C₃₇ ⋊ D₈₇₆
Sylow	C₃ ⋊ D₁₀₈₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₁₀₃
commutator, socle, cosocle	C₂×C₃×C₃₇×C₇₃	C₂²
Fitting	C₃×C₄×C₃₇×C₇₃	C₂
hypercenter	C₄	D₈₁₀₃
hypocenter	C₃×C₃₇×C₇₃	D₄
LCS	G ▷ C₂×C₃×C₃₇×C₇₃ ▷ C₃×C₃₇×C₇₃
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 117
transitive degree	μ_t(G) = 32412

A minimal faithful permutation representation — 2 generators on 117 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);(2,4)(6,7)(9,44)(10,43)(11,42)(12,41)(13,40)(14,39)(15,38)(16,37)(17,36)(18,35)(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2^3,3,37,73|b^2=c,b^a=bc,d^a=d^2,e^a=e^36,f^a=f^72

Conjugacy classes & character table

16209 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, 37, 73]);;
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.5^36);;
gap> SetConjugate(coll, 6, 1, F.6^72);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32412 #72302122