D₂₆₄₀₄ #72289756

order	52808
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₆₄₀₄

Splittings

Fitting	C₄×C₇×C₂₃×C₄₁ ⋊ C₂
Hall	C₇×C₂₃×C₄₁ ⋊ D₄
Hall	C₂₃×C₄₁ ⋊ D₂₈
Hall	C₇×C₄₁ ⋊ D₉₂
Hall	C₇×C₂₃ ⋊ D₁₆₄
Sylow	C₄₁ ⋊ D₆₄₄
Sylow	C₂₃ ⋊ D₁₁₄₈
Sylow	C₇ ⋊ D₃₇₇₂

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₆₆₀₁
commutator, socle, cosocle	C₂×C₇×C₂₃×C₄₁	C₂²
Fitting	C₄×C₇×C₂₃×C₄₁	C₂
hypercenter	C₄	D₆₆₀₁
hypocenter	C₇×C₂₃×C₄₁	D₄
LCS	G ▷ C₂×C₇×C₂₃×C₄₁ ▷ C₇×C₂₃×C₄₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 75
transitive degree	μ_t(G) = 26404

A minimal faithful permutation representation — 2 generators on 75 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);(2,4)(6,11)(7,10)(8,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,75)(37,74)(38,73)(39,72)(40,71)(41,70)(42,69)(43,68)(44,67)(45,66)(46,65)(47,64)(48,63)(49,62)(50,61)(51,60)(52,59)(53,58)(54,57)(55,56)

Polycyclic presentation:

2^3,7,23,41|b^2=c,b^a=bc,d^a=d^6,e^a=e^22,f^a=f^40

Conjugacy classes & character table

13205 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, 23, 41]);;
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.5^22);;
gap> SetConjugate(coll, 6, 1, F.6^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D26404 #72289756