D₁₅₉₇₅ #44544006

order	31950
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₉₇₅
unambiguous semidirect product	C₉⋊D₁₇₇₅
unambiguous semidirect product	C₂₅⋊D₆₃₉
unambiguous semidirect product	C₇₁⋊D₂₂₅

Splittings

Fitting, derived, Hall	C₉×C₂₅×C₇₁ ⋊ C₂
Hall	C₂₅×C₇₁ ⋊ D₉
Hall	C₉×C₇₁ ⋊ D₂₅
Hall	C₉×C₂₅ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₂₂₅
Sylow	C₂₅ ⋊ D₆₃₉
Sylow	C₉ ⋊ D₁₇₇₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₅×C₇₁	C₂
Frattini	C₃×C₅	D₁₀₆₅
socle	C₃×C₅×C₇₁	D₁₅
LCS	G ▷ C₉×C₂₅×C₇₁
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 105
transitive degree	μ_t(G) = 15975

A minimal faithful permutation representation — 2 generators on 105 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);(2,9)(3,8)(4,7)(5,6)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(36,105)(37,104)(38,103)(39,102)(40,101)(41,100)(42,99)(43,98)(44,97)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2,3,5,71,3,5|b^3=e^2,c^5=f^4,b^a=b^2e,c^a=c^4f,d^a=d^70,e^a=e^2,f^a=f^4

Conjugacy classes & character table

7989 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₉₇₅
maximal normal subgroup of	C₂×D₁₅₉₇₅
maximal quotient of	Dic₁₅₉₇₅, C₂×D₁₅₉₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31950, 9);

D15975 #44544006