Finite Group Databasebeta

← Prev (#11745445) Next (#11745447) →

D₄₆₈ #11745446

order	936
automorphism	Aut(G) order 67392
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₄₆₈
unambiguous semidirect product	C₄⋊D₁₁₇

Splittings

Fitting	C₄×C₉×C₁₃ ⋊ C₂
Hall	C₉×C₁₃ ⋊ D₄
Sylow	C₁₃ ⋊ D₃₆
Sylow	C₉ ⋊ D₅₂

Subgroups

	subgroup	G/H
center	C₂	C₂×D₁₁₇
commutator, cosocle	C₂×C₉×C₁₃	C₂²
Frattini	C₂×C₃	C₂×D₃₉
Fitting	C₄×C₉×C₁₃	C₂
socle	C₂×C₃×C₁₃	C₂×D₃
hypercenter	C₄	D₁₁₇
hypocenter	C₉×C₁₃	D₄
LCS	G ▷ C₂×C₉×C₁₃ ▷ C₉×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 468

A minimal faithful permutation representation — 2 generators on 26 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);(2,4)(6,13)(7,12)(8,11)(9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)

Polycyclic presentation:

2^3,3,13,3|b^2=c,d^3=f,b^a=bc,d^a=d^2f^2,e^a=e^12,f^a=f^2

Conjugacy classes & character table

237 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#14272403, #14272404, #14272409, …
maximal normal subgroup of	#14272403, #14272404, #14272409, …
maximal quotient of	Dic₄₆₈, #14272468, D₉₃₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(936, 39);