D₃₂₃₉₆ #72302113

order	64792
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₃₂₃₉₆

Splittings

Fitting	C₄×C₇×C₁₃×C₈₉ ⋊ C₂
Hall	C₇×C₁₃×C₈₉ ⋊ D₄
Hall	C₁₃×C₈₉ ⋊ D₂₈
Hall	C₇×C₈₉ ⋊ D₅₂
Hall	C₇×C₁₃ ⋊ D₃₅₆
Sylow	C₈₉ ⋊ D₃₆₄
Sylow	C₁₃ ⋊ D₂₄₉₂
Sylow	C₇ ⋊ D₄₆₂₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₈₀₉₉
commutator, socle, cosocle	C₂×C₇×C₁₃×C₈₉	C₂²
Fitting	C₄×C₇×C₁₃×C₈₉	C₂
hypercenter	C₄	D₈₀₉₉
hypocenter	C₇×C₁₃×C₈₉	D₄
LCS	G ▷ C₂×C₇×C₁₃×C₈₉ ▷ C₇×C₁₃×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 32396

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

Polycyclic presentation:

2^3,7,13,89|b^2=c,b^a=bc,d^a=d^6,e^a=e^12,f^a=f^88

Conjugacy classes & character table

16201 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 7, 13, 89]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^6);;
gap> SetConjugate(coll, 5, 1, F.5^12);;
gap> SetConjugate(coll, 6, 1, F.6^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D32396 #72302113