D₃₀₉₆₉ #72297903

order	61938
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₉₆₉

Splittings

Fitting, derived, Hall	C₂₇×C₃₁×C₃₇ ⋊ C₂
Hall	C₃₁×C₃₇ ⋊ D₂₇
Hall	C₂₇×C₃₇ ⋊ D₃₁
Hall	C₂₇×C₃₁ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₈₃₇
Sylow	C₃₁ ⋊ D₉₉₉
Sylow	C₂₇ ⋊ D₁₁₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₇×C₃₁×C₃₇	C₂
Frattini	C₉	D₃₄₄₁
socle	C₃×C₃₁×C₃₇	D₉
LCS	G ▷ C₂₇×C₃₁×C₃₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 30969

A minimal faithful permutation representation — 2 generators on 95 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);(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(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:

2,3,31,37,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^30,d^a=d^36,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

15486 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, 31, 37, 3, 3]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2^2*F.5^2*F.6^2);;
gap> SetConjugate(coll, 3, 1, F.3^30);;
gap> SetConjugate(coll, 4, 1, F.4^36);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D30969 #72297903