D₈₈₃₅ #31798968

order	17670
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₈₃₅
unambiguous semidirect product	C₃⋊D₂₉₄₅
unambiguous semidirect product	C₅⋊D₁₇₆₇
unambiguous semidirect product	C₁₉⋊D₄₆₅
unambiguous semidirect product	C₃₁⋊D₂₈₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₁₉×C₃₁ ⋊ C₂
Hall	C₅×C₁₉×C₃₁ ⋊ D₃
Hall	C₃×C₁₉×C₃₁ ⋊ D₅
Hall	C₁₉×C₃₁ ⋊ D₁₅
Hall	C₃×C₅×C₃₁ ⋊ D₁₉
Hall	C₃×C₅×C₁₉ ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ D₅₇
Hall	C₅×C₁₉ ⋊ D₉₃
Hall	C₃×C₃₁ ⋊ D₉₅
Hall	C₃×C₁₉ ⋊ D₁₅₅
Sylow	C₃₁ ⋊ D₂₈₅
Sylow	C₁₉ ⋊ D₄₆₅
Hall	C₃×C₅ ⋊ D₅₈₉
Sylow	C₅ ⋊ D₁₇₆₇
Sylow	C₃ ⋊ D₂₉₄₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₁₉×C₃₁	C₂
LCS	G ▷ C₃×C₅×C₁₉×C₃₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 8835

A minimal faithful permutation representation — 2 generators on 58 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);(2,3)(5,8)(6,7)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(29,58)(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:

2,3,5,19,31|b^a=b^2,c^a=c^4,d^a=d^18,e^a=e^30

Conjugacy classes & character table

4419 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₂₈₅, D₁₉×D₄₆₅, D₁₅×D₅₈₉, …
maximal normal subgroup of	D₃₁×D₂₈₅, D₁₉×D₄₆₅, D₁₅×D₅₈₉, …
maximal quotient of	Dic₈₈₃₅, C₂×D₈₈₃₅, D₂₆₅₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(17670, 25);

D8835 #31798968