D₃₂₂₃₂ #72301974

order	64464
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₂₃₂

Splittings

Fitting	C₃×C₈×C₁₇×C₇₉ ⋊ C₂
Hall	C₃×C₁₇×C₇₉ ⋊ D₈
Hall	C₁₇×C₇₉ ⋊ D₂₄
Hall	C₃×C₇₉ ⋊ D₁₃₆
Sylow	C₇₉ ⋊ D₄₀₈
Hall	C₃×C₁₇ ⋊ D₆₃₂
Sylow	C₁₇ ⋊ D₁₈₉₆
Sylow	C₃ ⋊ D₁₀₇₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₆₁₁₆
commutator, cosocle	C₃×C₄×C₁₇×C₇₉	C₂²
Frattini	C₄	C₂×D₄₀₂₉
Fitting	C₃×C₈×C₁₇×C₇₉	C₂
socle	C₂×C₃×C₁₇×C₇₉	D₄
hypercenter	C₈	D₄₀₂₉
hypocenter	C₃×C₁₇×C₇₉	D₈
LCS	G ▷ C₃×C₄×C₁₇×C₇₉ ▷ C₂×C₃×C₁₇×C₇₉ ▷ C₃×C₁₇×C₇₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 32232

A minimal faithful permutation representation — 2 generators on 107 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);(2,8)(3,7)(4,6)(10,11)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,107)(31,106)(32,105)(33,104)(34,103)(35,102)(36,101)(37,100)(38,99)(39,98)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^4,3,17,79|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^16,g^a=g^78

Conjugacy classes & character table

16119 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, 3, 17, 79]);;
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.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^16);;
gap> SetConjugate(coll, 7, 1, F.7^78);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32232 #72301974