D₁₆₃₅₃ #45658406

order	32706
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₃₅₃
unambiguous semidirect product	C₉⋊D₁₈₁₇
unambiguous semidirect product	C₂₃⋊D₇₁₁
unambiguous semidirect product	C₇₉⋊D₂₀₇

Splittings

Fitting, derived, Hall	C₉×C₂₃×C₇₉ ⋊ C₂
Hall	C₂₃×C₇₉ ⋊ D₉
Hall	C₉×C₇₉ ⋊ D₂₃
Hall	C₉×C₂₃ ⋊ D₇₉
Sylow	C₇₉ ⋊ D₂₀₇
Sylow	C₂₃ ⋊ D₇₁₁
Sylow	C₉ ⋊ D₁₈₁₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₃×C₇₉	C₂
Frattini	C₃	D₅₄₅₁
socle	C₃×C₂₃×C₇₉	D₃
LCS	G ▷ C₉×C₂₃×C₇₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 16353

A minimal faithful permutation representation — 2 generators on 111 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);(2,9)(3,8)(4,7)(5,6)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2,3,23,79,3|b^3=e^2,b^a=b^2e,c^a=c^22,d^a=d^78,e^a=e^2

Conjugacy classes & character table

8178 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₃₅₃
maximal normal subgroup of	C₂×D₁₆₃₅₃
maximal quotient of	Dic₁₆₃₅₃, C₂×D₁₆₃₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32706, 9);

D16353 #45658406