D₁₆₆₇₅ #45688643

order	33350
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₆₇₅
unambiguous semidirect product	C₂₃⋊D₇₂₅
unambiguous semidirect product	C₂₅⋊D₆₆₇
unambiguous semidirect product	C₂₉⋊D₅₇₅

Splittings

Fitting, derived, Hall	C₂₃×C₂₅×C₂₉ ⋊ C₂
Hall	C₂₅×C₂₉ ⋊ D₂₃
Hall	C₂₃×C₂₉ ⋊ D₂₅
Hall	C₂₃×C₂₅ ⋊ D₂₉
Sylow	C₂₉ ⋊ D₅₇₅
Sylow	C₂₅ ⋊ D₆₆₇
Sylow	C₂₃ ⋊ D₇₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₃×C₂₅×C₂₉	C₂
Frattini	C₅	D₃₃₃₅
socle	C₅×C₂₃×C₂₉	D₅
LCS	G ▷ C₂₃×C₂₅×C₂₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 77
transitive degree	μ_t(G) = 16675

A minimal faithful permutation representation — 2 generators on 77 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);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2,5,23,29,5|b^5=e^4,b^a=b^4e,c^a=c^22,d^a=d^28,e^a=e^4

Conjugacy classes & character table

8339 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33350, 8);

D16675 #45688643