D₁₂₂₄₃ #37587501

order	24486
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₂₄₃
unambiguous semidirect product	C₃⋊D₄₀₈₁
unambiguous semidirect product	C₇⋊D₁₇₄₉
unambiguous semidirect product	C₁₁⋊D₁₁₁₃
unambiguous semidirect product	C₅₃⋊D₂₃₁

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₁×C₅₃ ⋊ C₂
Hall	C₇×C₁₁×C₅₃ ⋊ D₃
Hall	C₃×C₁₁×C₅₃ ⋊ D₇
Hall	C₃×C₇×C₅₃ ⋊ D₁₁
Hall	C₁₁×C₅₃ ⋊ D₂₁
Hall	C₇×C₅₃ ⋊ D₃₃
Hall	C₃×C₇×C₁₁ ⋊ D₅₃
Hall	C₃×C₅₃ ⋊ D₇₇
Hall	C₇×C₁₁ ⋊ D₁₅₉
Sylow	C₅₃ ⋊ D₂₃₁
Hall	C₃×C₁₁ ⋊ D₃₇₁
Hall	C₃×C₇ ⋊ D₅₈₃
Sylow	C₁₁ ⋊ D₁₁₁₃
Sylow	C₇ ⋊ D₁₇₄₉
Sylow	C₃ ⋊ D₄₀₈₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₁×C₅₃	C₂
LCS	G ▷ C₃×C₇×C₁₁×C₅₃
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 12243

A minimal faithful permutation representation — 2 generators on 74 points:

(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74);(2,3)(5,10)(6,9)(7,8)(12,21)(13,20)(14,19)(15,18)(16,17)(23,74)(24,73)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,3,7,11,53|b^a=b^2,c^a=c^6,d^a=d^10,e^a=e^52

Conjugacy classes & character table

6123 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24486, 17);

D12243 #37587501