D₁₅₃₃₆ #72272052

order	30672
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₅₃₃₆

Splittings

Fitting	C₈×C₂₇×C₇₁ ⋊ C₂
Hall	C₂₇×C₇₁ ⋊ D₈
Sylow	C₇₁ ⋊ D₂₁₆
Sylow	C₂₇ ⋊ D₅₆₈

Subgroups

	subgroup	G/H
center	C₂	D₇₆₆₈
commutator, cosocle	C₄×C₂₇×C₇₁	C₂²
Frattini	C₄×C₉	C₂×D₂₁₃
Fitting	C₈×C₂₇×C₇₁	C₂
socle	C₂×C₃×C₇₁	D₃₆
hypercenter	C₈	D₁₉₁₇
hypocenter	C₂₇×C₇₁	D₈
LCS	G ▷ C₄×C₂₇×C₇₁ ▷ C₂×C₂₇×C₇₁ ▷ C₂₇×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 106
transitive degree	μ_t(G) = 15336

A minimal faithful permutation representation — 2 generators on 106 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);(2,8)(3,7)(4,6)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(37,106)(38,105)(39,104)(40,103)(41,102)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2^4,3,71,3^2|b^2=cd,c^2=d,e^3=g,g^3=h,b^a=bc,c^a=cd,e^a=e^2g^2h^2,f^a=f^70,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

7671 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₀₆₇₂
maximal normal subgroup of	D₃₀₆₇₂
maximal quotient of	Dic₁₅₃₃₆, D₃₀₆₇₂

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", "g", "h");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 71, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D15336 #72272052