D₁₂₁₀₃ #37564141

order	24206
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₂₁₀₃
unambiguous semidirect product	C₁₃⋊D₉₃₁
unambiguous semidirect product	C₁₉⋊D₆₃₇
unambiguous semidirect product	C₄₉⋊D₂₄₇

Splittings

Fitting, derived, Hall	C₁₃×C₁₉×C₄₉ ⋊ C₂
Hall	C₁₉×C₄₉ ⋊ D₁₃
Hall	C₁₃×C₄₉ ⋊ D₁₉
Hall	C₁₃×C₁₉ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₂₄₇
Sylow	C₁₉ ⋊ D₆₃₇
Sylow	C₁₃ ⋊ D₉₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₃×C₁₉×C₄₉	C₂
Frattini	C₇	D₁₇₂₉
socle	C₇×C₁₃×C₁₉	D₇
LCS	G ▷ C₁₃×C₁₉×C₄₉
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 12103

A minimal faithful permutation representation — 2 generators on 81 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)(63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,62)(52,61)(53,60)(54,59)(55,58)(56,57)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

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

Conjugacy classes & character table

6053 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24206, 8);

D12103 #37564141