D₁₁₁₉₃ #36375599

order	22386
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₁₉₃
unambiguous semidirect product	C₃⋊D₃₇₃₁
unambiguous semidirect product	C₇⋊D₁₅₉₉
unambiguous semidirect product	C₁₃⋊D₈₆₁
unambiguous semidirect product	C₄₁⋊D₂₇₃

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₃×C₄₁ ⋊ C₂
Hall	C₇×C₁₃×C₄₁ ⋊ D₃
Hall	C₃×C₁₃×C₄₁ ⋊ D₇
Hall	C₃×C₇×C₄₁ ⋊ D₁₃
Hall	C₁₃×C₄₁ ⋊ D₂₁
Hall	C₇×C₄₁ ⋊ D₃₉
Hall	C₃×C₇×C₁₃ ⋊ D₄₁
Hall	C₃×C₄₁ ⋊ D₉₁
Hall	C₇×C₁₃ ⋊ D₁₂₃
Sylow	C₄₁ ⋊ D₂₇₃
Hall	C₃×C₁₃ ⋊ D₂₈₇
Hall	C₃×C₇ ⋊ D₅₃₃
Sylow	C₁₃ ⋊ D₈₆₁
Sylow	C₇ ⋊ D₁₅₉₉
Sylow	C₃ ⋊ D₃₇₃₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₃×C₄₁	C₂
LCS	G ▷ C₃×C₇×C₁₃×C₄₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 11193

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21,22,23)(24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64);(2,3)(5,10)(6,9)(7,8)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Polycyclic presentation:

2,3,7,13,41|b^a=b^2,c^a=c^6,d^a=d^12,e^a=e^40

Conjugacy classes & character table

5598 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22386, 20);

D11193 #36375599