D₁₁₇₁₆ #72223946

order	23432
automorphism	Aut(G) order 65609600
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₁₇₁₆

Splittings

Fitting	C₄×C₂₉×C₁₀₁ ⋊ C₂
Hall	C₂₉×C₁₀₁ ⋊ D₄
Sylow	C₁₀₁ ⋊ D₁₁₆
Sylow	C₂₉ ⋊ D₄₀₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₉₂₉
commutator, socle, cosocle	C₂×C₂₉×C₁₀₁	C₂²
Fitting	C₄×C₂₉×C₁₀₁	C₂
hypercenter	C₄	D₂₉₂₉
hypocenter	C₂₉×C₁₀₁	D₄
LCS	G ▷ C₂×C₂₉×C₁₀₁ ▷ C₂₉×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 134
transitive degree	μ_t(G) = 11716

A minimal faithful permutation representation — 2 generators on 134 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)(34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134);(2,4)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(35,134)(36,133)(37,132)(38,131)(39,130)(40,129)(41,128)(42,127)(43,126)(44,125)(45,124)(46,123)(47,122)(48,121)(49,120)(50,119)(51,118)(52,117)(53,116)(54,115)(55,114)(56,113)(57,112)(58,111)(59,110)(60,109)(61,108)(62,107)(63,106)(64,105)(65,104)(66,103)(67,102)(68,101)(69,100)(70,99)(71,98)(72,97)(73,96)(74,95)(75,94)(76,93)(77,92)(78,91)(79,90)(80,89)(81,88)(82,87)(83,86)(84,85)

Polycyclic presentation:

2^3,29,101|b^2=c,b^a=bc,d^a=d^28,e^a=e^100

Conjugacy classes & character table

5861 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃₄₃₂
maximal normal subgroup of	D₂₃₄₃₂
maximal quotient of	Dic₁₁₇₁₆, D₂₃₄₃₂

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, 29, 101]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D11716 #72223946