D₃₀₇₅₃ #72296136

order	61506
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral

D₃₀₇₅₃

Splittings

Fitting, derived, Hall	C₁₇×C₂₇×C₆₇ ⋊ C₂
Hall	C₂₇×C₆₇ ⋊ D₁₇
Hall	C₁₇×C₆₇ ⋊ D₂₇
Hall	C₁₇×C₂₇ ⋊ D₆₇
Sylow	C₆₇ ⋊ D₄₅₉
Sylow	C₂₇ ⋊ D₁₁₃₉
Sylow	C₁₇ ⋊ D₁₈₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₇×C₂₇×C₆₇	C₂
Frattini	C₉	D₃₄₁₇
socle	C₃×C₁₇×C₆₇	D₉
LCS	G ▷ C₁₇×C₂₇×C₆₇
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 30753

A minimal faithful permutation representation — 2 generators on 111 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);(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)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2,3,17,67,3^2|b^3=e,e^3=f,b^a=b^2e^2f^2,c^a=c^16,d^a=d^66,e^a=e^2f^2,f^a=f^2

Conjugacy classes & character table

15378 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, 17, 67, 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.3^16);;
gap> SetConjugate(coll, 4, 1, F.4^66);;
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);

D30753 #72296136