D₂₇₄₆₈ #72290771

order	54936
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₇₄₆₈

Splittings

Fitting	C₄×C₇×C₉×C₁₀₉ ⋊ C₂
Hall	C₇×C₉×C₁₀₉ ⋊ D₄
Hall	C₉×C₁₀₉ ⋊ D₂₈
Hall	C₇×C₁₀₉ ⋊ D₃₆
Sylow	C₁₀₉ ⋊ D₂₅₂
Hall	C₇×C₉ ⋊ D₄₃₆
Sylow	C₉ ⋊ D₃₀₅₂
Sylow	C₇ ⋊ D₃₉₂₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₆₈₆₇
commutator, cosocle	C₂×C₇×C₉×C₁₀₉	C₂²
Frattini	C₂×C₃	C₂×D₂₂₈₉
Fitting	C₄×C₇×C₉×C₁₀₉	C₂
socle	C₂×C₃×C₇×C₁₀₉	C₂×D₃
hypercenter	C₄	D₆₈₆₇
hypocenter	C₇×C₉×C₁₀₉	D₄
LCS	G ▷ C₂×C₇×C₉×C₁₀₉ ▷ C₇×C₉×C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 27468

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

Polycyclic presentation:

2^3,3,7,109,3|b^2=c,d^3=g,b^a=bc,d^a=d^2g^2,e^a=e^6,f^a=f^108,g^a=g^2

Conjugacy classes & character table

13737 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", "g");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 7, 109, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^108);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D27468 #72290771