D₁₁₀₆₇ #36348995

order	22134
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₀₆₇
unambiguous semidirect product	C₃⋊D₃₆₈₉
unambiguous semidirect product	C₇⋊D₁₅₈₁
unambiguous semidirect product	C₁₇⋊D₆₅₁
unambiguous semidirect product	C₃₁⋊D₃₅₇

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₇×C₃₁ ⋊ C₂
Hall	C₇×C₁₇×C₃₁ ⋊ D₃
Hall	C₃×C₁₇×C₃₁ ⋊ D₇
Hall	C₃×C₇×C₃₁ ⋊ D₁₇
Hall	C₁₇×C₃₁ ⋊ D₂₁
Hall	C₃×C₇×C₁₇ ⋊ D₃₁
Hall	C₇×C₃₁ ⋊ D₅₁
Hall	C₇×C₁₇ ⋊ D₉₃
Hall	C₃×C₃₁ ⋊ D₁₁₉
Hall	C₃×C₁₇ ⋊ D₂₁₇
Sylow	C₃₁ ⋊ D₃₅₇
Hall	C₃×C₇ ⋊ D₅₂₇
Sylow	C₁₇ ⋊ D₆₅₁
Sylow	C₇ ⋊ D₁₅₈₁
Sylow	C₃ ⋊ D₃₆₈₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₇×C₃₁	C₂
LCS	G ▷ C₃×C₇×C₁₇×C₃₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 58
transitive degree	μ_t(G) = 11067

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)(5,10)(6,9)(7,8)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(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:

2,3,7,17,31|b^a=b^2,c^a=c^6,d^a=d^16,e^a=e^30

Conjugacy classes & character table

5535 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22134, 20);

D11067 #36348995