D₂₀₇₅ #16861908

order	4150
automorphism	Aut(G) order 3403000
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₂₀₇₅
unambiguous semidirect product	C₂₅⋊D₈₃
unambiguous semidirect product	C₈₃⋊D₂₅

Splittings

Fitting, derived, Hall	C₂₅×C₈₃ ⋊ C₂
Sylow	C₈₃ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₈₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₅×C₈₃	C₂
Frattini	C₅	D₄₁₅
socle	C₅×C₈₃	D₅
LCS	G ▷ C₂₅×C₈₃
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 108
transitive degree	μ_t(G) = 2075

A minimal faithful permutation representation — 2 generators on 108 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,108)(28,107)(29,106)(30,105)(31,104)(32,103)(33,102)(34,101)(35,100)(36,99)(37,98)(38,97)(39,96)(40,95)(41,94)(42,93)(43,92)(44,91)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2,5,83,5|b^5=d^4,b^a=b^4d,c^a=c^82,d^a=d^4

Conjugacy classes & character table

1039 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₂₀₇₅, D₆₂₂₅, C₇×D₂₀₇₅, …
maximal normal subgroup of	C₃×D₂₀₇₅, C₇×D₂₀₇₅
maximal quotient of	C₃×D₂₀₇₅, D₆₂₂₅, C₇×D₂₀₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4150, 4);

D2075 #16861908