D₈₃₀₈ #72213232

order	16616
automorphism	Aut(G) order 32899680
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₈₃₀₈

Splittings

Fitting	C₄×C₃₁×C₆₇ ⋊ C₂
Hall	C₃₁×C₆₇ ⋊ D₄
Sylow	C₆₇ ⋊ D₁₂₄
Sylow	C₃₁ ⋊ D₂₆₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₀₇₇
commutator, socle, cosocle	C₂×C₃₁×C₆₇	C₂²
Fitting	C₄×C₃₁×C₆₇	C₂
hypercenter	C₄	D₂₀₇₇
hypocenter	C₃₁×C₆₇	D₄
LCS	G ▷ C₂×C₃₁×C₆₇ ▷ C₃₁×C₆₇
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 8308

A minimal faithful permutation representation — 2 generators on 102 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);(2,4)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(37,102)(38,101)(39,100)(40,99)(41,98)(42,97)(43,96)(44,95)(45,94)(46,93)(47,92)(48,91)(49,90)(50,89)(51,88)(52,87)(53,86)(54,85)(55,84)(56,83)(57,82)(58,81)(59,80)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74)(66,73)(67,72)(68,71)(69,70)

Polycyclic presentation:

2^3,31,67|b^2=c,b^a=bc,d^a=d^30,e^a=e^66

Conjugacy classes & character table

4157 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₆₆₁₆, D₂₄₉₂₄
maximal normal subgroup of	D₁₆₆₁₆
maximal quotient of	Dic₈₃₀₈, D₁₆₆₁₆, D₂₄₉₂₄

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 2, 31, 67]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^66);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D8308 #72213232