D₅₄₂₃ #24925734

order	10846
automorphism	Aut(G) order 24295040
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₄₂₃
unambiguous semidirect product	C₁₁⋊D₄₉₃
unambiguous semidirect product	C₁₇⋊D₃₁₉
unambiguous semidirect product	C₂₉⋊D₁₈₇

Splittings

Fitting, socle, derived, Hall	C₁₁×C₁₇×C₂₉ ⋊ C₂
Hall	C₁₇×C₂₉ ⋊ D₁₁
Hall	C₁₁×C₂₉ ⋊ D₁₇
Hall	C₁₁×C₁₇ ⋊ D₂₉
Sylow	C₂₉ ⋊ D₁₈₇
Sylow	C₁₇ ⋊ D₃₁₉
Sylow	C₁₁ ⋊ D₄₉₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₁₇×C₂₉	C₂
LCS	G ▷ C₁₁×C₁₇×C₂₉
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 5423

A minimal faithful permutation representation — 2 generators on 57 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2,11,17,29|b^a=b^10,c^a=c^16,d^a=d^28

Conjugacy classes & character table

2713 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₉×D₁₈₇, D₁₇×D₃₁₉, D₁₁×D₄₉₃, …
maximal normal subgroup of	D₂₉×D₁₈₇, D₁₇×D₃₁₉, D₁₁×D₄₉₃, …
maximal quotient of	Dic₅₄₂₃, C₂×D₅₄₂₃, C₃×D₅₄₂₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10846, 8);

D5423 #24925734