D₆₆₉₃ #27211468

order	13386
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₆₉₃
unambiguous semidirect product	C₃⋊D₂₂₃₁
unambiguous semidirect product	C₂₃⋊D₂₉₁
unambiguous semidirect product	C₉₇⋊D₆₉

Splittings

Fitting, socle, derived, Hall	C₃×C₂₃×C₉₇ ⋊ C₂
Hall	C₂₃×C₉₇ ⋊ D₃
Hall	C₃×C₉₇ ⋊ D₂₃
Sylow	C₉₇ ⋊ D₆₉
Hall	C₃×C₂₃ ⋊ D₉₇
Sylow	C₂₃ ⋊ D₂₉₁
Sylow	C₃ ⋊ D₂₂₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₂₃×C₉₇	C₂
LCS	G ▷ C₃×C₂₃×C₉₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 6693

A minimal faithful permutation representation — 2 generators on 123 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,3)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(28,123)(29,122)(30,121)(31,120)(32,119)(33,118)(34,117)(35,116)(36,115)(37,114)(38,113)(39,112)(40,111)(41,110)(42,109)(43,108)(44,107)(45,106)(46,105)(47,104)(48,103)(49,102)(50,101)(51,100)(52,99)(53,98)(54,97)(55,96)(56,95)(57,94)(58,93)(59,92)(60,91)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)

Polycyclic presentation:

2,3,23,97|b^a=b^2,c^a=c^22,d^a=d^96

Conjugacy classes & character table

3348 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₉×D₉₇, D₂₃×D₂₉₁, D₃×D₂₂₃₁, …
maximal normal subgroup of	D₆₉×D₉₇, D₂₃×D₂₉₁, D₃×D₂₂₃₁, …
maximal quotient of	Dic₆₆₉₃, C₂×D₆₆₉₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13386, 9);

D6693 #27211468