D₁₅₅₆₁ #44527546

order	31122
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₅₆₁
unambiguous semidirect product	C₇⋊D₂₂₂₃
unambiguous semidirect product	C₉⋊D₁₇₂₉
unambiguous semidirect product	C₁₃⋊D₁₁₉₇
unambiguous semidirect product	C₁₉⋊D₈₁₉

Splittings

Fitting, derived, Hall	C₇×C₉×C₁₃×C₁₉ ⋊ C₂
Hall	C₉×C₁₃×C₁₉ ⋊ D₇
Hall	C₇×C₁₃×C₁₉ ⋊ D₉
Hall	C₇×C₉×C₁₉ ⋊ D₁₃
Hall	C₇×C₉×C₁₃ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ D₆₃
Hall	C₉×C₁₉ ⋊ D₉₁
Hall	C₇×C₁₉ ⋊ D₁₁₇
Hall	C₉×C₁₃ ⋊ D₁₃₃
Hall	C₇×C₁₃ ⋊ D₁₇₁
Hall	C₇×C₉ ⋊ D₂₄₇
Sylow	C₁₉ ⋊ D₈₁₉
Sylow	C₁₃ ⋊ D₁₁₉₇
Sylow	C₉ ⋊ D₁₇₂₉
Sylow	C₇ ⋊ D₂₂₂₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₉×C₁₃×C₁₉	C₂
Frattini	C₃	D₅₁₈₇
socle	C₃×C₇×C₁₃×C₁₉	D₃
LCS	G ▷ C₇×C₉×C₁₃×C₁₉
UCS	C₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 15561

A minimal faithful permutation representation — 2 generators on 48 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);(2,9)(3,8)(4,7)(5,6)(11,16)(12,15)(13,14)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)

Polycyclic presentation:

2,3,7,13,19,3|b^3=f^2,b^a=b^2f,c^a=c^6,d^a=d^12,e^a=e^18,f^a=f^2

Conjugacy classes & character table

7782 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₅₆₁
maximal normal subgroup of	C₂×D₁₅₅₆₁
maximal quotient of	Dic₁₅₅₆₁, C₂×D₁₅₅₆₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31122, 29);

D15561 #44527546