D₂₇₆₇₆ #72290989

order	55352
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₆₇₆

Splittings

Fitting	C₄×C₁₁×C₁₇×C₃₇ ⋊ C₂
Hall	C₁₁×C₁₇×C₃₇ ⋊ D₄
Hall	C₁₇×C₃₇ ⋊ D₄₄
Hall	C₁₁×C₃₇ ⋊ D₆₈
Hall	C₁₁×C₁₇ ⋊ D₁₄₈
Sylow	C₃₇ ⋊ D₇₄₈
Sylow	C₁₇ ⋊ D₁₆₂₈
Sylow	C₁₁ ⋊ D₂₅₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₉₁₉
commutator, socle, cosocle	C₂×C₁₁×C₁₇×C₃₇	C₂²
Fitting	C₄×C₁₁×C₁₇×C₃₇	C₂
hypercenter	C₄	D₆₉₁₉
hypocenter	C₁₁×C₁₇×C₃₇	D₄
LCS	G ▷ C₂×C₁₁×C₁₇×C₃₇ ▷ C₁₁×C₁₇×C₃₇
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 27676

A minimal faithful permutation representation — 2 generators on 69 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);(2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2^3,11,17,37|b^2=c,b^a=bc,d^a=d^10,e^a=e^16,f^a=f^36

Conjugacy classes & character table

13841 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, 11, 17, 37]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^10);;
gap> SetConjugate(coll, 5, 1, F.5^16);;
gap> SetConjugate(coll, 6, 1, F.6^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27676 #72290989