D₉₁₇₇ #32913465

order	18354
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₁₇₇
unambiguous semidirect product	C₃⋊D₃₀₅₉
unambiguous semidirect product	C₇⋊D₁₃₁₁
unambiguous semidirect product	C₁₉⋊D₄₈₃
unambiguous semidirect product	C₂₃⋊D₃₉₉

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₉×C₂₃ ⋊ C₂
Hall	C₇×C₁₉×C₂₃ ⋊ D₃
Hall	C₃×C₁₉×C₂₃ ⋊ D₇
Hall	C₃×C₇×C₂₃ ⋊ D₁₉
Hall	C₁₉×C₂₃ ⋊ D₂₁
Hall	C₃×C₇×C₁₉ ⋊ D₂₃
Hall	C₇×C₂₃ ⋊ D₅₇
Hall	C₇×C₁₉ ⋊ D₆₉
Hall	C₃×C₂₃ ⋊ D₁₃₃
Hall	C₃×C₁₉ ⋊ D₁₆₁
Sylow	C₂₃ ⋊ D₃₉₉
Hall	C₃×C₇ ⋊ D₄₃₇
Sylow	C₁₉ ⋊ D₄₈₃
Sylow	C₇ ⋊ D₁₃₁₁
Sylow	C₃ ⋊ D₃₀₅₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₉×C₂₃	C₂
LCS	G ▷ C₃×C₇×C₁₉×C₂₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 9177

A minimal faithful permutation representation — 2 generators on 52 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);(2,3)(5,10)(6,9)(7,8)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(31,52)(32,51)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(41,42)

Polycyclic presentation:

2,3,7,19,23|b^a=b^2,c^a=c^6,d^a=d^18,e^a=e^22

Conjugacy classes & character table

4590 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₃×D₃₉₉, D₁₉×D₄₈₃, D₂₁×D₄₃₇, …
maximal normal subgroup of	D₂₃×D₃₉₉, D₁₉×D₄₈₃, D₂₁×D₄₃₇, …
maximal quotient of	Dic₉₁₇₇, C₂×D₉₁₇₇, D₂₇₅₃₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18354, 20);

D9177 #32913465