D₁₆₇₄₄ #72272829

order	33488
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₁₆₇₄₄

Splittings

Fitting	C₇×C₈×C₁₃×C₂₃ ⋊ C₂
Hall	C₇×C₁₃×C₂₃ ⋊ D₈
Hall	C₁₃×C₂₃ ⋊ D₅₆
Hall	C₇×C₂₃ ⋊ D₁₀₄
Hall	C₇×C₁₃ ⋊ D₁₈₄
Sylow	C₂₃ ⋊ D₇₂₈
Sylow	C₁₃ ⋊ D₁₂₈₈
Sylow	C₇ ⋊ D₂₃₉₂

Subgroups

	subgroup	G/H
center	C₂	D₈₃₇₂
commutator, cosocle	C₄×C₇×C₁₃×C₂₃	C₂²
Frattini	C₄	C₂×D₂₀₉₃
Fitting	C₇×C₈×C₁₃×C₂₃	C₂
socle	C₂×C₇×C₁₃×C₂₃	D₄
hypercenter	C₈	D₂₀₉₃
hypocenter	C₇×C₁₃×C₂₃	D₈
LCS	G ▷ C₄×C₇×C₁₃×C₂₃ ▷ C₂×C₇×C₁₃×C₂₃ ▷ C₇×C₁₃×C₂₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 16744

A minimal faithful permutation representation — 2 generators on 51 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);(2,8)(3,7)(4,6)(10,15)(11,14)(12,13)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(30,51)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)

Polycyclic presentation:

2^4,7,13,23|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^6,f^a=f^12,g^a=g^22

Conjugacy classes & character table

8375 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, 2, 7, 13, 23]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^6);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^22);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D16744 #72272829